let o, m be set ; :: thesis: c1Cat o,m is Category-like
set C = c1Cat o,m;
set CP = the Comp of (c1Cat o,m);
set CD = the Source of (c1Cat o,m);
set CC = the Target of (c1Cat o,m);
set CI = the Id of (c1Cat o,m);
thus
for f, g being Morphism of (c1Cat o,m) holds
( [g,f] in dom the Comp of (c1Cat o,m) iff the Source of (c1Cat o,m) . g = the Target of (c1Cat o,m) . f )
:: according to CAT_1:def 8 :: thesis: ( ( for b1, b2 being Element of the carrier' of (c1Cat o,m) holds
( not the Source of (c1Cat o,m) . b2 = the Target of (c1Cat o,m) . b1 or ( the Source of (c1Cat o,m) . (the Comp of (c1Cat o,m) . b2,b1) = the Source of (c1Cat o,m) . b1 & the Target of (c1Cat o,m) . (the Comp of (c1Cat o,m) . b2,b1) = the Target of (c1Cat o,m) . b2 ) ) ) & ( for b1, b2, b3 being Element of the carrier' of (c1Cat o,m) holds
( not the Source of (c1Cat o,m) . b3 = the Target of (c1Cat o,m) . b2 or not the Source of (c1Cat o,m) . b2 = the Target of (c1Cat o,m) . b1 or the Comp of (c1Cat o,m) . b3,(the Comp of (c1Cat o,m) . b2,b1) = the Comp of (c1Cat o,m) . (the Comp of (c1Cat o,m) . b3,b2),b1 ) ) & ( for b1 being Element of the carrier of (c1Cat o,m) holds
( the Source of (c1Cat o,m) . (the Id of (c1Cat o,m) . b1) = b1 & the Target of (c1Cat o,m) . (the Id of (c1Cat o,m) . b1) = b1 & ( for b2 being Element of the carrier' of (c1Cat o,m) holds
( not the Target of (c1Cat o,m) . b2 = b1 or the Comp of (c1Cat o,m) . (the Id of (c1Cat o,m) . b1),b2 = b2 ) ) & ( for b2 being Element of the carrier' of (c1Cat o,m) holds
( not the Source of (c1Cat o,m) . b2 = b1 or the Comp of (c1Cat o,m) . b2,(the Id of (c1Cat o,m) . b1) = b2 ) ) ) ) )proof
let f,
g be
Morphism of
(c1Cat o,m);
:: thesis: ( [g,f] in dom the Comp of (c1Cat o,m) iff the Source of (c1Cat o,m) . g = the Target of (c1Cat o,m) . f )
(
f = m &
g = m &
dom the
Comp of
(c1Cat o,m) = {[m,m]} )
by FUNCOP_1:19, TARSKI:def 1;
hence
(
[g,f] in dom the
Comp of
(c1Cat o,m) iff the
Source of
(c1Cat o,m) . g = the
Target of
(c1Cat o,m) . f )
by TARSKI:def 1;
:: thesis: verum
end;
thus
for f, g being Morphism of (c1Cat o,m) st the Source of (c1Cat o,m) . g = the Target of (c1Cat o,m) . f holds
( the Source of (c1Cat o,m) . (the Comp of (c1Cat o,m) . g,f) = the Source of (c1Cat o,m) . f & the Target of (c1Cat o,m) . (the Comp of (c1Cat o,m) . g,f) = the Target of (c1Cat o,m) . g )
:: thesis: ( ( for b1, b2, b3 being Element of the carrier' of (c1Cat o,m) holds
( not the Source of (c1Cat o,m) . b3 = the Target of (c1Cat o,m) . b2 or not the Source of (c1Cat o,m) . b2 = the Target of (c1Cat o,m) . b1 or the Comp of (c1Cat o,m) . b3,(the Comp of (c1Cat o,m) . b2,b1) = the Comp of (c1Cat o,m) . (the Comp of (c1Cat o,m) . b3,b2),b1 ) ) & ( for b1 being Element of the carrier of (c1Cat o,m) holds
( the Source of (c1Cat o,m) . (the Id of (c1Cat o,m) . b1) = b1 & the Target of (c1Cat o,m) . (the Id of (c1Cat o,m) . b1) = b1 & ( for b2 being Element of the carrier' of (c1Cat o,m) holds
( not the Target of (c1Cat o,m) . b2 = b1 or the Comp of (c1Cat o,m) . (the Id of (c1Cat o,m) . b1),b2 = b2 ) ) & ( for b2 being Element of the carrier' of (c1Cat o,m) holds
( not the Source of (c1Cat o,m) . b2 = b1 or the Comp of (c1Cat o,m) . b2,(the Id of (c1Cat o,m) . b1) = b2 ) ) ) ) )proof
let f,
g be
Morphism of
(c1Cat o,m);
:: thesis: ( the Source of (c1Cat o,m) . g = the Target of (c1Cat o,m) . f implies ( the Source of (c1Cat o,m) . (the Comp of (c1Cat o,m) . g,f) = the Source of (c1Cat o,m) . f & the Target of (c1Cat o,m) . (the Comp of (c1Cat o,m) . g,f) = the Target of (c1Cat o,m) . g ) )
the
Comp of
(c1Cat o,m) . g,
f = m
by FUNCOP_1:92;
then reconsider gf = the
Comp of
(c1Cat o,m) . g,
f as
Morphism of
(c1Cat o,m) by TARSKI:def 1;
( the
Source of
(c1Cat o,m) . f = o & the
Target of
(c1Cat o,m) . g = o & the
Source of
(c1Cat o,m) . gf = o & the
Target of
(c1Cat o,m) . gf = o )
by FUNCT_2:65;
hence
( the
Source of
(c1Cat o,m) . g = the
Target of
(c1Cat o,m) . f implies ( the
Source of
(c1Cat o,m) . (the Comp of (c1Cat o,m) . g,f) = the
Source of
(c1Cat o,m) . f & the
Target of
(c1Cat o,m) . (the Comp of (c1Cat o,m) . g,f) = the
Target of
(c1Cat o,m) . g ) )
;
:: thesis: verum
end;
thus
for f, g, h being Morphism of (c1Cat o,m) st the Source of (c1Cat o,m) . h = the Target of (c1Cat o,m) . g & the Source of (c1Cat o,m) . g = the Target of (c1Cat o,m) . f holds
the Comp of (c1Cat o,m) . h,(the Comp of (c1Cat o,m) . g,f) = the Comp of (c1Cat o,m) . (the Comp of (c1Cat o,m) . h,g),f
:: thesis: for b1 being Element of the carrier of (c1Cat o,m) holds
( the Source of (c1Cat o,m) . (the Id of (c1Cat o,m) . b1) = b1 & the Target of (c1Cat o,m) . (the Id of (c1Cat o,m) . b1) = b1 & ( for b2 being Element of the carrier' of (c1Cat o,m) holds
( not the Target of (c1Cat o,m) . b2 = b1 or the Comp of (c1Cat o,m) . (the Id of (c1Cat o,m) . b1),b2 = b2 ) ) & ( for b2 being Element of the carrier' of (c1Cat o,m) holds
( not the Source of (c1Cat o,m) . b2 = b1 or the Comp of (c1Cat o,m) . b2,(the Id of (c1Cat o,m) . b1) = b2 ) ) )proof
let f,
g,
h be
Morphism of
(c1Cat o,m);
:: thesis: ( the Source of (c1Cat o,m) . h = the Target of (c1Cat o,m) . g & the Source of (c1Cat o,m) . g = the Target of (c1Cat o,m) . f implies the Comp of (c1Cat o,m) . h,(the Comp of (c1Cat o,m) . g,f) = the Comp of (c1Cat o,m) . (the Comp of (c1Cat o,m) . h,g),f )
( the
Comp of
(c1Cat o,m) . g,
f = m & the
Comp of
(c1Cat o,m) . h,
g = m )
by FUNCOP_1:92;
then reconsider gf = the
Comp of
(c1Cat o,m) . g,
f,
hg = the
Comp of
(c1Cat o,m) . h,
g as
Morphism of
(c1Cat o,m) by TARSKI:def 1;
( the
Comp of
(c1Cat o,m) . h,
gf = m & the
Comp of
(c1Cat o,m) . hg,
f = m )
by FUNCOP_1:92;
hence
( the
Source of
(c1Cat o,m) . h = the
Target of
(c1Cat o,m) . g & the
Source of
(c1Cat o,m) . g = the
Target of
(c1Cat o,m) . f implies the
Comp of
(c1Cat o,m) . h,
(the Comp of (c1Cat o,m) . g,f) = the
Comp of
(c1Cat o,m) . (the Comp of (c1Cat o,m) . h,g),
f )
;
:: thesis: verum
end;
let b be Object of (c1Cat o,m); :: thesis: ( the Source of (c1Cat o,m) . (the Id of (c1Cat o,m) . b) = b & the Target of (c1Cat o,m) . (the Id of (c1Cat o,m) . b) = b & ( for b1 being Element of the carrier' of (c1Cat o,m) holds
( not the Target of (c1Cat o,m) . b1 = b or the Comp of (c1Cat o,m) . (the Id of (c1Cat o,m) . b),b1 = b1 ) ) & ( for b1 being Element of the carrier' of (c1Cat o,m) holds
( not the Source of (c1Cat o,m) . b1 = b or the Comp of (c1Cat o,m) . b1,(the Id of (c1Cat o,m) . b) = b1 ) ) )
b = o
by TARSKI:def 1;
hence
( the Source of (c1Cat o,m) . (the Id of (c1Cat o,m) . b) = b & the Target of (c1Cat o,m) . (the Id of (c1Cat o,m) . b) = b )
by FUNCT_2:65; :: thesis: ( ( for b1 being Element of the carrier' of (c1Cat o,m) holds
( not the Target of (c1Cat o,m) . b1 = b or the Comp of (c1Cat o,m) . (the Id of (c1Cat o,m) . b),b1 = b1 ) ) & ( for b1 being Element of the carrier' of (c1Cat o,m) holds
( not the Source of (c1Cat o,m) . b1 = b or the Comp of (c1Cat o,m) . b1,(the Id of (c1Cat o,m) . b) = b1 ) ) )
thus
for f being Morphism of (c1Cat o,m) st the Target of (c1Cat o,m) . f = b holds
the Comp of (c1Cat o,m) . (the Id of (c1Cat o,m) . b),f = f
:: thesis: for b1 being Element of the carrier' of (c1Cat o,m) holds
( not the Source of (c1Cat o,m) . b1 = b or the Comp of (c1Cat o,m) . b1,(the Id of (c1Cat o,m) . b) = b1 )
let g be Morphism of (c1Cat o,m); :: thesis: ( not the Source of (c1Cat o,m) . g = b or the Comp of (c1Cat o,m) . g,(the Id of (c1Cat o,m) . b) = g )
assume
the Source of (c1Cat o,m) . g = b
; :: thesis: the Comp of (c1Cat o,m) . g,(the Id of (c1Cat o,m) . b) = g
g = m
by TARSKI:def 1;
hence
the Comp of (c1Cat o,m) . g,(the Id of (c1Cat o,m) . b) = g
by FUNCOP_1:92; :: thesis: verum