Theorem cati 15102

Description: Definitional properties of a category.

Hypotheses

Ref	Expression
cati.1	|- D = (dom` T)
cati.2	|- C = (cod` T)
cati.3	|- J = (id` T)
cati.4	|- R = (o` T)
cati.5	|- M = dom D
cati.6	|- O = dom J

Assertion

Ref	Expression
cati	|- (T e. Cat -> ((<.<.D, C>., <.J, R>.>. e. Ded /\ A.f e. M A.g e. M A.h e. M (((D` h) = (C` g) /\ (D` g) = (C` f)) -> (hR(gRf)) = ((hRg)Rf))) /\ (A.a e. O A.f e. M ((C` f) = a -> ((J` a)Rf) = f) /\ A.a e. O A.f e. M ((D` f) = a -> (fR(J` a)) = f))))

Distinct variable groups:   C,a,f,g,h   D,a,f,g,h   J,a,f   f,M   R,a,f,g,h

Proof of Theorem cati

Step	Hyp	Ref	Expression
1		df-cat 15100	. . 3 |- Cat = {x | E.dE.cE.jE.r(x = <.<.d, c>., <.j, r>.>. /\ ((<.<.d, c>., <.j, r>.>. e. Ded /\ A.f e. dom dA.g e. dom dA.h e. dom d(((d` h) = (c` g) /\ (d` g) = (c` f)) -> (hr(grf)) = ((hrg)rf))) /\ (A.a e. dom jA.f e. dom d((c` f) = a -> ((j` a)rf) = f) /\ A.a e. dom jA.f e. dom d((d` f) = a -> (fr(j` a)) = f))))}
2	1	eleq2i 1961	. 2 |- (T e. Cat <-> T e. {x | E.dE.cE.jE.r(x = <.<.d, c>., <.j, r>.>. /\ ((<.<.d, c>., <.j, r>.>. e. Ded /\ A.f e. dom dA.g e. dom dA.h e. dom d(((d` h) = (c` g) /\ (d` g) = (c` f)) -> (hr(grf)) = ((hrg)rf))) /\ (A.a e. dom jA.f e. dom d((c` f) = a -> ((j` a)rf) = f) /\ A.a e. dom jA.f e. dom d((d` f) = a -> (fr(j` a)) = f))))})
3		cati.1	. . . . . . 7 |- D = (dom` T)
4	3	domval 15070	. . . . . 6 |- D = (1st` (1st` T))
5	4	eqcomi 1888	. . . . 5 |- (1st` (1st` T)) = D
6	5	eqeq2i 1894	. . . 4 |- (d = (1st` (1st` T)) <-> d = D)
7		opeq1 3158	. . . . . . . 8 |- (d = D -> <.d, c>. = <.D, c>.)
8	7	opeq1d 3164	. . . . . . 7 |- (d = D -> <.<.d, c>., <.j, r>.>. = <.<.D, c>., <.j, r>.>.)
9	8	eleq1d 1963	. . . . . 6 |- (d = D -> (<.<.d, c>., <.j, r>.>. e. Ded <-> <.<.D, c>., <.j, r>.>. e. Ded ))
10		dmeq 4157	. . . . . . . . . 10 |- (d = D -> dom d = dom D)
11		cati.5	. . . . . . . . . 10 |- M = dom D
12	10, 11	syl6eqr 1946	. . . . . . . . 9 |- (d = D -> dom d = M)
13	12	eleq2d 1964	. . . . . . . 8 |- (d = D -> (f e. dom d <-> f e. M))
14	12	eleq2d 1964	. . . . . . . . . 10 |- (d = D -> (g e. dom d <-> g e. M))
15	12	eleq2d 1964	. . . . . . . . . . . 12 |- (d = D -> (h e. dom d <-> h e. M))
16		fveq1 4680	. . . . . . . . . . . . . . 15 |- (d = D -> (d` h) = (D` h))
17	16	eqeq1d 1892	. . . . . . . . . . . . . 14 |- (d = D -> ((d` h) = (c` g) <-> (D` h) = (c` g)))
18		fveq1 4680	. . . . . . . . . . . . . . 15 |- (d = D -> (d` g) = (D` g))
19	18	eqeq1d 1892	. . . . . . . . . . . . . 14 |- (d = D -> ((d` g) = (c` f) <-> (D` g) = (c` f)))
20	17, 19	anbi12d 690	. . . . . . . . . . . . 13 |- (d = D -> (((d` h) = (c` g) /\ (d` g) = (c` f)) <-> ((D` h) = (c` g) /\ (D` g) = (c` f))))
21	20	imbi1d 675	. . . . . . . . . . . 12 |- (d = D -> ((((d` h) = (c` g) /\ (d` g) = (c` f)) -> (hr(grf)) = ((hrg)rf)) <-> (((D` h) = (c` g) /\ (D` g) = (c` f)) -> (hr(grf)) = ((hrg)rf))))
22	15, 21	imbi12d 688	. . . . . . . . . . 11 |- (d = D -> ((h e. dom d -> (((d` h) = (c` g) /\ (d` g) = (c` f)) -> (hr(grf)) = ((hrg)rf))) <-> (h e. M -> (((D` h) = (c` g) /\ (D` g) = (c` f)) -> (hr(grf)) = ((hrg)rf)))))
23	22	ralbidv2 2125	. . . . . . . . . 10 |- (d = D -> (A.h e. dom d(((d` h) = (c` g) /\ (d` g) = (c` f)) -> (hr(grf)) = ((hrg)rf)) <-> A.h e. M (((D` h) = (c` g) /\ (D` g) = (c` f)) -> (hr(grf)) = ((hrg)rf))))
24	14, 23	imbi12d 688	. . . . . . . . 9 |- (d = D -> ((g e. dom d -> A.h e. dom d(((d` h) = (c` g) /\ (d` g) = (c` f)) -> (hr(grf)) = ((hrg)rf))) <-> (g e. M -> A.h e. M (((D` h) = (c` g) /\ (D` g) = (c` f)) -> (hr(grf)) = ((hrg)rf)))))
25	24	ralbidv2 2125	. . . . . . . 8 |- (d = D -> (A.g e. dom dA.h e. dom d(((d` h) = (c` g) /\ (d` g) = (c` f)) -> (hr(grf)) = ((hrg)rf)) <-> A.g e. M A.h e. M (((D` h) = (c` g) /\ (D` g) = (c` f)) -> (hr(grf)) = ((hrg)rf))))
26	13, 25	imbi12d 688	. . . . . . 7 |- (d = D -> ((f e. dom d -> A.g e. dom dA.h e. dom d(((d` h) = (c` g) /\ (d` g) = (c` f)) -> (hr(grf)) = ((hrg)rf))) <-> (f e. M -> A.g e. M A.h e. M (((D` h) = (c` g) /\ (D` g) = (c` f)) -> (hr(grf)) = ((hrg)rf)))))
27	26	ralbidv2 2125	. . . . . 6 |- (d = D -> (A.f e. dom dA.g e. dom dA.h e. dom d(((d` h) = (c` g) /\ (d` g) = (c` f)) -> (hr(grf)) = ((hrg)rf)) <-> A.f e. M A.g e. M A.h e. M (((D` h) = (c` g) /\ (D` g) = (c` f)) -> (hr(grf)) = ((hrg)rf))))
28	9, 27	anbi12d 690	. . . . 5 |- (d = D -> ((<.<.d, c>., <.j, r>.>. e. Ded /\ A.f e. dom dA.g e. dom dA.h e. dom d(((d` h) = (c` g) /\ (d` g) = (c` f)) -> (hr(grf)) = ((hrg)rf))) <-> (<.<.D, c>., <.j, r>.>. e. Ded /\ A.f e. M A.g e. M A.h e. M (((D` h) = (c` g) /\ (D` g) = (c` f)) -> (hr(grf)) = ((hrg)rf)))))
29	12	raleqdv 2269	. . . . . . 7 |- (d = D -> (A.f e. dom d((c` f) = a -> ((j` a)rf) = f) <-> A.f e. M ((c` f) = a -> ((j` a)rf) = f)))
30	29	ralbidv 2123	. . . . . 6 |- (d = D -> (A.a e. dom jA.f e. dom d((c` f) = a -> ((j` a)rf) = f) <-> A.a e. dom jA.f e. M ((c` f) = a -> ((j` a)rf) = f)))
31		fveq1 4680	. . . . . . . . . . 11 |- (d = D -> (d` f) = (D` f))
32	31	eqeq1d 1892	. . . . . . . . . 10 |- (d = D -> ((d` f) = a <-> (D` f) = a))
33	32	imbi1d 675	. . . . . . . . 9 |- (d = D -> (((d` f) = a -> (fr(j` a)) = f) <-> ((D` f) = a -> (fr(j` a)) = f)))
34	13, 33	imbi12d 688	. . . . . . . 8 |- (d = D -> ((f e. dom d -> ((d` f) = a -> (fr(j` a)) = f)) <-> (f e. M -> ((D` f) = a -> (fr(j` a)) = f))))
35	34	ralbidv2 2125	. . . . . . 7 |- (d = D -> (A.f e. dom d((d` f) = a -> (fr(j` a)) = f) <-> A.f e. M ((D` f) = a -> (fr(j` a)) = f)))
36	35	ralbidv 2123	. . . . . 6 |- (d = D -> (A.a e. dom jA.f e. dom d((d` f) = a -> (fr(j` a)) = f) <-> A.a e. dom jA.f e. M ((D` f) = a -> (fr(j` a)) = f)))
37	30, 36	anbi12d 690	. . . . 5 |- (d = D -> ((A.a e. dom jA.f e. dom d((c` f) = a -> ((j` a)rf) = f) /\ A.a e. dom jA.f e. dom d((d` f) = a -> (fr(j` a)) = f)) <-> (A.a e. dom jA.f e. M ((c` f) = a -> ((j` a)rf) = f) /\ A.a e. dom jA.f e. M ((D` f) = a -> (fr(j` a)) = f))))
38	28, 37	anbi12d 690	. . . 4 |- (d = D -> (((<.<.d, c>., <.j, r>.>. e. Ded /\ A.f e. dom dA.g e. dom dA.h e. dom d(((d` h) = (c` g) /\ (d` g) = (c` f)) -> (hr(grf)) = ((hrg)rf))) /\ (A.a e. dom jA.f e. dom d((c` f) = a -> ((j` a)rf) = f) /\ A.a e. dom jA.f e. dom d((d` f) = a -> (fr(j` a)) = f))) <-> ((<.<.D, c>., <.j, r>.>. e. Ded /\ A.f e. M A.g e. M A.h e. M (((D` h) = (c` g) /\ (D` g) = (c` f)) -> (hr(grf)) = ((hrg)rf))) /\ (A.a e. dom jA.f e. M ((c` f) = a -> ((j` a)rf) = f) /\ A.a e. dom jA.f e. M ((D` f) = a -> (fr(j` a)) = f)))))
39	6, 38	sylbi 216	. . 3 |- (d = (1st` (1st` T)) -> (((<.<.d, c>., <.j, r>.>. e. Ded /\ A.f e. dom dA.g e. dom dA.h e. dom d(((d` h) = (c` g) /\ (d` g) = (c` f)) -> (hr(grf)) = ((hrg)rf))) /\ (A.a e. dom jA.f e. dom d((c` f) = a -> ((j` a)rf) = f) /\ A.a e. dom jA.f e. dom d((d` f) = a -> (fr(j` a)) = f))) <-> ((<.<.D, c>., <.j, r>.>. e. Ded /\ A.f e. M A.g e. M A.h e. M (((D` h) = (c` g) /\ (D` g) = (c` f)) -> (hr(grf)) = ((hrg)rf))) /\ (A.a e. dom jA.f e. M ((c` f) = a -> ((j` a)rf) = f) /\ A.a e. dom jA.f e. M ((D` f) = a -> (fr(j` a)) = f)))))
40		cati.2	. . . . . . 7 |- C = (cod` T)
41	40	codval 15071	. . . . . 6 |- C = (2nd` (1st` T))
42	41	eqcomi 1888	. . . . 5 |- (2nd` (1st` T)) = C
43	42	eqeq2i 1894	. . . 4 |- (c = (2nd` (1st` T)) <-> c = C)
44		opeq2 3159	. . . . . . . 8 |- (c = C -> <.D, c>. = <.D, C>.)
45	44	opeq1d 3164	. . . . . . 7 |- (c = C -> <.<.D, c>., <.j, r>.>. = <.<.D, C>., <.j, r>.>.)
46	45	eleq1d 1963	. . . . . 6 |- (c = C -> (<.<.D, c>., <.j, r>.>. e. Ded <-> <.<.D, C>., <.j, r>.>. e. Ded ))
47		fveq1 4680	. . . . . . . . . . 11 |- (c = C -> (c` g) = (C` g))
48	47	eqeq2d 1895	. . . . . . . . . 10 |- (c = C -> ((D` h) = (c` g) <-> (D` h) = (C` g)))
49		fveq1 4680	. . . . . . . . . . 11 |- (c = C -> (c` f) = (C` f))
50	49	eqeq2d 1895	. . . . . . . . . 10 |- (c = C -> ((D` g) = (c` f) <-> (D` g) = (C` f)))
51	48, 50	anbi12d 690	. . . . . . . . 9 |- (c = C -> (((D` h) = (c` g) /\ (D` g) = (c` f)) <-> ((D` h) = (C` g) /\ (D` g) = (C` f))))
52	51	imbi1d 675	. . . . . . . 8 |- (c = C -> ((((D` h) = (c` g) /\ (D` g) = (c` f)) -> (hr(grf)) = ((hrg)rf)) <-> (((D` h) = (C` g) /\ (D` g) = (C` f)) -> (hr(grf)) = ((hrg)rf))))
53	52	ralbidv 2123	. . . . . . 7 |- (c = C -> (A.h e. M (((D` h) = (c` g) /\ (D` g) = (c` f)) -> (hr(grf)) = ((hrg)rf)) <-> A.h e. M (((D` h) = (C` g) /\ (D` g) = (C` f)) -> (hr(grf)) = ((hrg)rf))))
54	53	2ralbidv 2140	. . . . . 6 |- (c = C -> (A.f e. M A.g e. M A.h e. M (((D` h) = (c` g) /\ (D` g) = (c` f)) -> (hr(grf)) = ((hrg)rf)) <-> A.f e. M A.g e. M A.h e. M (((D` h) = (C` g) /\ (D` g) = (C` f)) -> (hr(grf)) = ((hrg)rf))))
55	46, 54	anbi12d 690	. . . . 5 |- (c = C -> ((<.<.D, c>., <.j, r>.>. e. Ded /\ A.f e. M A.g e. M A.h e. M (((D` h) = (c` g) /\ (D` g) = (c` f)) -> (hr(grf)) = ((hrg)rf))) <-> (<.<.D, C>., <.j, r>.>. e. Ded /\ A.f e. M A.g e. M A.h e. M (((D` h) = (C` g) /\ (D` g) = (C` f)) -> (hr(grf)) = ((hrg)rf)))))
56	49	eqeq1d 1892	. . . . . . . 8 |- (c = C -> ((c` f) = a <-> (C` f) = a))
57	56	imbi1d 675	. . . . . . 7 |- (c = C -> (((c` f) = a -> ((j` a)rf) = f) <-> ((C` f) = a -> ((j` a)rf) = f)))
58	57	2ralbidv 2140	. . . . . 6 |- (c = C -> (A.a e. dom jA.f e. M ((c` f) = a -> ((j` a)rf) = f) <-> A.a e. dom jA.f e. M ((C` f) = a -> ((j` a)rf) = f)))
59	58	anbi1d 679	. . . . 5 |- (c = C -> ((A.a e. dom jA.f e. M ((c` f) = a -> ((j` a)rf) = f) /\ A.a e. dom jA.f e. M ((D` f) = a -> (fr(j` a)) = f)) <-> (A.a e. dom jA.f e. M ((C` f) = a -> ((j` a)rf) = f) /\ A.a e. dom jA.f e. M ((D` f) = a -> (fr(j` a)) = f))))
60	55, 59	anbi12d 690	. . . 4 |- (c = C -> (((<.<.D, c>., <.j, r>.>. e. Ded /\ A.f e. M A.g e. M A.h e. M (((D` h) = (c` g) /\ (D` g) = (c` f)) -> (hr(grf)) = ((hrg)rf))) /\ (A.a e. dom jA.f e. M ((c` f) = a -> ((j` a)rf) = f) /\ A.a e. dom jA.f e. M ((D` f) = a -> (fr(j` a)) = f))) <-> ((<.<.D, C>., <.j, r>.>. e. Ded /\ A.f e. M A.g e. M A.h e. M (((D` h) = (C` g) /\ (D` g) = (C` f)) -> (hr(grf)) = ((hrg)rf))) /\ (A.a e. dom jA.f e. M ((C` f) = a -> ((j` a)rf) = f) /\ A.a e. dom jA.f e. M ((D` f) = a -> (fr(j` a)) = f)))))
61	43, 60	sylbi 216	. . 3 |- (c = (2nd` (1st` T)) -> (((<.<.D, c>., <.j, r>.>. e. Ded /\ A.f e. M A.g e. M A.h e. M (((D` h) = (c` g) /\ (D` g) = (c` f)) -> (hr(grf)) = ((hrg)rf))) /\ (A.a e. dom jA.f e. M ((c` f) = a -> ((j` a)rf) = f) /\ A.a e. dom jA.f e. M ((D` f) = a -> (fr(j` a)) = f))) <-> ((<.<.D, C>., <.j, r>.>. e. Ded /\ A.f e. M A.g e. M A.h e. M (((D` h) = (C` g) /\ (D` g) = (C` f)) -> (hr(grf)) = ((hrg)rf))) /\ (A.a e. dom jA.f e. M ((C` f) = a -> ((j` a)rf) = f) /\ A.a e. dom jA.f e. M ((D` f) = a -> (fr(j` a)) = f)))))
62		cati.3	. . . . . . 7 |- J = (id` T)
63	62	idval 15072	. . . . . 6 |- J = (1st` (2nd` T))
64	63	eqcomi 1888	. . . . 5 |- (1st` (2nd` T)) = J
65	64	eqeq2i 1894	. . . 4 |- (j = (1st` (2nd` T)) <-> j = J)
66		opeq1 3158	. . . . . . . 8 |- (j = J -> <.j, r>. = <.J, r>.)
67	66	opeq2d 3165	. . . . . . 7 |- (j = J -> <.<.D, C>., <.j, r>.>. = <.<.D, C>., <.J, r>.>.)
68	67	eleq1d 1963	. . . . . 6 |- (j = J -> (<.<.D, C>., <.j, r>.>. e. Ded <-> <.<.D, C>., <.J, r>.>. e. Ded ))
69	68	anbi1d 679	. . . . 5 |- (j = J -> ((<.<.D, C>., <.j, r>.>. e. Ded /\ A.f e. M A.g e. M A.h e. M (((D` h) = (C` g) /\ (D` g) = (C` f)) -> (hr(grf)) = ((hrg)rf))) <-> (<.<.D, C>., <.J, r>.>. e. Ded /\ A.f e. M A.g e. M A.h e. M (((D` h) = (C` g) /\ (D` g) = (C` f)) -> (hr(grf)) = ((hrg)rf)))))
70		dmeq 4157	. . . . . . . . . 10 |- (j = J -> dom j = dom J)
71		cati.6	. . . . . . . . . 10 |- O = dom J
72	70, 71	syl6eqr 1946	. . . . . . . . 9 |- (j = J -> dom j = O)
73	72	eleq2d 1964	. . . . . . . 8 |- (j = J -> (a e. dom j <-> a e. O))
74		fveq1 4680	. . . . . . . . . . . 12 |- (j = J -> (j` a) = (J` a))
75	74	opreq1d 4897	. . . . . . . . . . 11 |- (j = J -> ((j` a)rf) = ((J` a)rf))
76	75	eqeq1d 1892	. . . . . . . . . 10 |- (j = J -> (((j` a)rf) = f <-> ((J` a)rf) = f))
77	76	imbi2d 674	. . . . . . . . 9 |- (j = J -> (((C` f) = a -> ((j` a)rf) = f) <-> ((C` f) = a -> ((J` a)rf) = f)))
78	77	ralbidv 2123	. . . . . . . 8 |- (j = J -> (A.f e. M ((C` f) = a -> ((j` a)rf) = f) <-> A.f e. M ((C` f) = a -> ((J` a)rf) = f)))
79	73, 78	imbi12d 688	. . . . . . 7 |- (j = J -> ((a e. dom j -> A.f e. M ((C` f) = a -> ((j` a)rf) = f)) <-> (a e. O -> A.f e. M ((C` f) = a -> ((J` a)rf) = f))))
80	79	ralbidv2 2125	. . . . . 6 |- (j = J -> (A.a e. dom jA.f e. M ((C` f) = a -> ((j` a)rf) = f) <-> A.a e. O A.f e. M ((C` f) = a -> ((J` a)rf) = f)))
81	74	opreq2d 4898	. . . . . . . . . . 11 |- (j = J -> (fr(j` a)) = (fr(J` a)))
82	81	eqeq1d 1892	. . . . . . . . . 10 |- (j = J -> ((fr(j` a)) = f <-> (fr(J` a)) = f))
83	82	imbi2d 674	. . . . . . . . 9 |- (j = J -> (((D` f) = a -> (fr(j` a)) = f) <-> ((D` f) = a -> (fr(J` a)) = f)))
84	83	ralbidv 2123	. . . . . . . 8 |- (j = J -> (A.f e. M ((D` f) = a -> (fr(j` a)) = f) <-> A.f e. M ((D` f) = a -> (fr(J` a)) = f)))
85	73, 84	imbi12d 688	. . . . . . 7 |- (j = J -> ((a e. dom j -> A.f e. M ((D` f) = a -> (fr(j` a)) = f)) <-> (a e. O -> A.f e. M ((D` f) = a -> (fr(J` a)) = f))))
86	85	ralbidv2 2125	. . . . . 6 |- (j = J -> (A.a e. dom jA.f e. M ((D` f) = a -> (fr(j` a)) = f) <-> A.a e. O A.f e. M ((D` f) = a -> (fr(J` a)) = f)))
87	80, 86	anbi12d 690	. . . . 5 |- (j = J -> ((A.a e. dom jA.f e. M ((C` f) = a -> ((j` a)rf) = f) /\ A.a e. dom jA.f e. M ((D` f) = a -> (fr(j` a)) = f)) <-> (A.a e. O A.f e. M ((C` f) = a -> ((J` a)rf) = f) /\ A.a e. O A.f e. M ((D` f) = a -> (fr(J` a)) = f))))
88	69, 87	anbi12d 690	. . . 4 |- (j = J -> (((<.<.D, C>., <.j, r>.>. e. Ded /\ A.f e. M A.g e. M A.h e. M (((D` h) = (C` g) /\ (D` g) = (C` f)) -> (hr(grf)) = ((hrg)rf))) /\ (A.a e. dom jA.f e. M ((C` f) = a -> ((j` a)rf) = f) /\ A.a e. dom jA.f e. M ((D` f) = a -> (fr(j` a)) = f))) <-> ((<.<.D, C>., <.J, r>.>. e. Ded /\ A.f e. M A.g e. M A.h e. M (((D` h) = (C` g) /\ (D` g) = (C` f)) -> (hr(grf)) = ((hrg)rf))) /\ (A.a e. O A.f e. M ((C` f) = a -> ((J` a)rf) = f) /\ A.a e. O A.f e. M ((D` f) = a -> (fr(J` a)) = f)))))
89	65, 88	sylbi 216	. . 3 |- (j = (1st` (2nd` T)) -> (((<.<.D, C>., <.j, r>.>. e. Ded /\ A.f e. M A.g e. M A.h e. M (((D` h) = (C` g) /\ (D` g) = (C` f)) -> (hr(grf)) = ((hrg)rf))) /\ (A.a e. dom jA.f e. M ((C` f) = a -> ((j` a)rf) = f) /\ A.a e. dom jA.f e. M ((D` f) = a -> (fr(j` a)) = f))) <-> ((<.<.D, C>., <.J, r>.>. e. Ded /\ A.f e. M A.g e. M A.h e. M (((D` h) = (C` g) /\ (D` g) = (C` f)) -> (hr(grf)) = ((hrg)rf))) /\ (A.a e. O A.f e. M ((C` f) = a -> ((J` a)rf) = f) /\ A.a e. O A.f e. M ((D` f) = a -> (fr(J` a)) = f)))))
90		cati.4	. . . . . . 7 |- R = (o` T)
91	90	cmpval 15073	. . . . . 6 |- R = (2nd` (2nd` T))
92	91	eqcomi 1888	. . . . 5 |- (2nd` (2nd` T)) = R
93	92	eqeq2i 1894	. . . 4 |- (r = (2nd` (2nd` T)) <-> r = R)
94		opeq2 3159	. . . . . . . 8 |- (r = R -> <.J, r>. = <.J, R>.)
95	94	opeq2d 3165	. . . . . . 7 |- (r = R -> <.<.D, C>., <.J, r>.>. = <.<.D, C>., <.J, R>.>.)
96	95	eleq1d 1963	. . . . . 6 |- (r = R -> (<.<.D, C>., <.J, r>.>. e. Ded <-> <.<.D, C>., <.J, R>.>. e. Ded ))
97		opreq 4888	. . . . . . . . . . . 12 |- (r = R -> (grf) = (gRf))
98	97	opreq2d 4898	. . . . . . . . . . 11 |- (r = R -> (hr(grf)) = (hr(gRf)))
99		opreq 4888	. . . . . . . . . . 11 |- (r = R -> (hr(gRf)) = (hR(gRf)))
100	98, 99	eqtrd 1925	. . . . . . . . . 10 |- (r = R -> (hr(grf)) = (hR(gRf)))
101		opreq 4888	. . . . . . . . . . . 12 |- (r = R -> (hrg) = (hRg))
102	101	opreq1d 4897	. . . . . . . . . . 11 |- (r = R -> ((hrg)rf) = ((hRg)rf))
103		opreq 4888	. . . . . . . . . . 11 |- (r = R -> ((hRg)rf) = ((hRg)Rf))
104	102, 103	eqtrd 1925	. . . . . . . . . 10 |- (r = R -> ((hrg)rf) = ((hRg)Rf))
105	100, 104	eqeq12d 1899	. . . . . . . . 9 |- (r = R -> ((hr(grf)) = ((hrg)rf) <-> (hR(gRf)) = ((hRg)Rf)))
106	105	imbi2d 674	. . . . . . . 8 |- (r = R -> ((((D` h) = (C` g) /\ (D` g) = (C` f)) -> (hr(grf)) = ((hrg)rf)) <-> (((D` h) = (C` g) /\ (D` g) = (C` f)) -> (hR(gRf)) = ((hRg)Rf))))
107	106	ralbidv 2123	. . . . . . 7 |- (r = R -> (A.h e. M (((D` h) = (C` g) /\ (D` g) = (C` f)) -> (hr(grf)) = ((hrg)rf)) <-> A.h e. M (((D` h) = (C` g) /\ (D` g) = (C` f)) -> (hR(gRf)) = ((hRg)Rf))))
108	107	2ralbidv 2140	. . . . . 6 |- (r = R -> (A.f e. M A.g e. M A.h e. M (((D` h) = (C` g) /\ (D` g) = (C` f)) -> (hr(grf)) = ((hrg)rf)) <-> A.f e. M A.g e. M A.h e. M (((D` h) = (C` g) /\ (D` g) = (C` f)) -> (hR(gRf)) = ((hRg)Rf))))
109	96, 108	anbi12d 690	. . . . 5 |- (r = R -> ((<.<.D, C>., <.J, r>.>. e. Ded /\ A.f e. M A.g e. M A.h e. M (((D` h) = (C` g) /\ (D` g) = (C` f)) -> (hr(grf)) = ((hrg)rf))) <-> (<.<.D, C>., <.J, R>.>. e. Ded /\ A.f e. M A.g e. M A.h e. M (((D` h) = (C` g) /\ (D` g) = (C` f)) -> (hR(gRf)) = ((hRg)Rf)))))
110		opreq 4888	. . . . . . . . 9 |- (r = R -> ((J` a)rf) = ((J` a)Rf))
111	110	eqeq1d 1892	. . . . . . . 8 |- (r = R -> (((J` a)rf) = f <-> ((J` a)Rf) = f))
112	111	imbi2d 674	. . . . . . 7 |- (r = R -> (((C` f) = a -> ((J` a)rf) = f) <-> ((C` f) = a -> ((J` a)Rf) = f)))
113	112	2ralbidv 2140	. . . . . 6 |- (r = R -> (A.a e. O A.f e. M ((C` f) = a -> ((J` a)rf) = f) <-> A.a e. O A.f e. M ((C` f) = a -> ((J` a)Rf) = f)))
114		opreq 4888	. . . . . . . . 9 |- (r = R -> (fr(J` a)) = (fR(J` a)))
115	114	eqeq1d 1892	. . . . . . . 8 |- (r = R -> ((fr(J` a)) = f <-> (fR(J` a)) = f))
116	115	imbi2d 674	. . . . . . 7 |- (r = R -> (((D` f) = a -> (fr(J` a)) = f) <-> ((D` f) = a -> (fR(J` a)) = f)))
117	116	2ralbidv 2140	. . . . . 6 |- (r = R -> (A.a e. O A.f e. M ((D` f) = a -> (fr(J` a)) = f) <-> A.a e. O A.f e. M ((D` f) = a -> (fR(J` a)) = f)))
118	113, 117	anbi12d 690	. . . . 5 |- (r = R -> ((A.a e. O A.f e. M ((C` f) = a -> ((J` a)rf) = f) /\ A.a e. O A.f e. M ((D` f) = a -> (fr(J` a)) = f)) <-> (A.a e. O A.f e. M ((C` f) = a -> ((J` a)Rf) = f) /\ A.a e. O A.f e. M ((D` f) = a -> (fR(J` a)) = f))))
119	109, 118	anbi12d 690	. . . 4 |- (r = R -> (((<.<.D, C>., <.J, r>.>. e. Ded /\ A.f e. M A.g e. M A.h e. M (((D` h) = (C` g) /\ (D` g) = (C` f)) -> (hr(grf)) = ((hrg)rf))) /\ (A.a e. O A.f e. M ((C` f) = a -> ((J` a)rf) = f) /\ A.a e. O A.f e. M ((D` f) = a -> (fr(J` a)) = f))) <-> ((<.<.D, C>., <.J, R>.>. e. Ded /\ A.f e. M A.g e. M A.h e. M (((D` h) = (C` g) /\ (D` g) = (C` f)) -> (hR(gRf)) = ((hRg)Rf))) /\ (A.a e. O A.f e. M ((C` f) = a -> ((J` a)Rf) = f) /\ A.a e. O A.f e. M ((D` f) = a -> (fR(J` a)) = f)))))
120	93, 119	sylbi 216	. . 3 |- (r = (2nd` (2nd` T)) -> (((<.<.D, C>., <.J, r>.>. e. Ded /\ A.f e. M A.g e. M A.h e. M (((D` h) = (C` g) /\ (D` g) = (C` f)) -> (hr(grf)) = ((hrg)rf))) /\ (A.a e. O A.f e. M ((C` f) = a -> ((J` a)rf) = f) /\ A.a e. O A.f e. M ((D` f) = a -> (fr(J` a)) = f))) <-> ((<.<.D, C>., <.J, R>.>. e. Ded /\ A.f e. M A.g e. M A.h e. M (((D` h) = (C` g) /\ (D` g) = (C` f)) -> (hR(gRf)) = ((hRg)Rf))) /\ (A.a e. O A.f e. M ((C` f) = a -> ((J` a)Rf) = f) /\ A.a e. O A.f e. M ((D` f) = a -> (fR(J` a)) = f)))))
121	39, 61, 89, 120	eloi 14400	. 2 |- (T e. {x | E.dE.cE.jE.r(x = <.<.d, c>., <.j, r>.>. /\ ((<.<.d, c>., <.j, r>.>. e. Ded /\ A.f e. dom dA.g e. dom dA.h e. dom d(((d` h) = (c` g) /\ (d` g) = (c` f)) -> (hr(grf)) = ((hrg)rf))) /\ (A.a e. dom jA.f e. dom d((c` f) = a -> ((j` a)rf) = f) /\ A.a e. dom jA.f e. dom d((d` f) = a -> (fr(j` a)) = f))))} -> ((<.<.D, C>., <.J, R>.>. e. Ded /\ A.f e. M A.g e. M A.h e. M (((D` h) = (C` g) /\ (D` g) = (C` f)) -> (hR(gRf)) = ((hRg)Rf))) /\ (A.a e. O A.f e. M ((C` f) = a -> ((J` a)Rf) = f) /\ A.a e. O A.f e. M ((D` f) = a -> (fR(J` a)) = f))))
122	2, 121	sylbi 216	1 |- (T e. Cat -> ((<.<.D, C>., <.J, R>.>. e. Ded /\ A.f e. M A.g e. M A.h e. M (((D` h) = (C` g) /\ (D` g) = (C` f)) -> (hR(gRf)) = ((hRg)Rf))) /\ (A.a e. O A.f e. M ((C` f) = a -> ((J` a)Rf) = f) /\ A.a e. O A.f e. M ((D` f) = a -> (fR(J` a)) = f))))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871  A.wral 2105  <.cop 3046  dom cdm 3986  ` cfv 3998  (class class class)co 4884  1stc1st 5018  2ndc2nd 5019  domcdom_ 15059  codccod_ 15060  idcid_ 15061  oco_ 15062   Ded cded 15081   Cat ccat 15099

This theorem is referenced by:  catded 15111  cmpasso 15120  cmpida 15121  cmpidb 15122

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fo 4012  df-fv 4014  df-opr 4886  df-1st 5020  df-2nd 5021  df-doma 15064  df-coda 15065  df-ida 15066  df-cmpa 15067  df-cat 15100

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