Theorem catciso 16757

Description: A functor is an isomorphism of categories if and only if it is full and faithful, and is a bijection on the objects. Remark 3.28(2) in [Adamek] p. 34. (Contributed by Mario Carneiro, 29-Jan-2017.)

Hypotheses

Ref	Expression
catciso.c	|- C = (CatCat ` U )
catciso.b	|- B = ( Base ` C )
catciso.r	|- R = ( Base ` X )
catciso.s	|- S = ( Base ` Y )
catciso.u	|- ( ph -> U e. V )
catciso.x	|- ( ph -> X e. B )
catciso.y	|- ( ph -> Y e. B )
catciso.i	|- I = ( Iso ` C )

Assertion

Ref	Expression
catciso	|- ( ph -> ( F e. ( X I Y ) <-> ( F e. ( ( X Full Y ) i^i ( X Faith Y ) ) /\ ( 1st ` F ) : R -1-1-onto-> S ) ) )

Proof of Theorem catciso

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relfunc 16522	. . . . 5 |- Rel ( X Func Y )
2		catciso.b	. . . . . . . . . . . . . 14 |- B = ( Base ` C )
3		eqid 2622	. . . . . . . . . . . . . 14 |- (Inv ` C ) = (Inv ` C )
4		catciso.u	. . . . . . . . . . . . . . 15 |- ( ph -> U e. V )
5		catciso.c	. . . . . . . . . . . . . . . 16 |- C = (CatCat ` U )
6	5	catccat 16754	. . . . . . . . . . . . . . 15 |- ( U e. V -> C e. Cat )
7	4, 6	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> C e. Cat )
8		catciso.x	. . . . . . . . . . . . . 14 |- ( ph -> X e. B )
9		catciso.y	. . . . . . . . . . . . . 14 |- ( ph -> Y e. B )
10		catciso.i	. . . . . . . . . . . . . 14 |- I = ( Iso ` C )
11	2, 3, 7, 8, 9, 10	isoval 16425	. . . . . . . . . . . . 13 |- ( ph -> ( X I Y ) = dom ( X (Inv ` C ) Y ) )
12	11	eleq2d 2687	. . . . . . . . . . . 12 |- ( ph -> ( F e. ( X I Y ) <-> F e. dom ( X (Inv ` C ) Y ) ) )
13	12	biimpa 501	. . . . . . . . . . 11 |- ( ( ph /\ F e. ( X I Y ) ) -> F e. dom ( X (Inv ` C ) Y ) )
14	7	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ F e. ( X I Y ) ) -> C e. Cat )
15	8	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ F e. ( X I Y ) ) -> X e. B )
16	9	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ F e. ( X I Y ) ) -> Y e. B )
17	2, 3, 14, 15, 16	invfun 16424	. . . . . . . . . . . 12 |- ( ( ph /\ F e. ( X I Y ) ) -> Fun ( X (Inv ` C ) Y ) )
18		funfvbrb 6330	. . . . . . . . . . . 12 |- ( Fun ( X (Inv ` C ) Y ) -> ( F e. dom ( X (Inv ` C ) Y ) <-> F ( X (Inv ` C ) Y ) ( ( X (Inv ` C ) Y ) ` F ) ) )
19	17, 18	syl 17	. . . . . . . . . . 11 |- ( ( ph /\ F e. ( X I Y ) ) -> ( F e. dom ( X (Inv ` C ) Y ) <-> F ( X (Inv ` C ) Y ) ( ( X (Inv ` C ) Y ) ` F ) ) )
20	13, 19	mpbid 222	. . . . . . . . . 10 |- ( ( ph /\ F e. ( X I Y ) ) -> F ( X (Inv ` C ) Y ) ( ( X (Inv ` C ) Y ) ` F ) )
21		eqid 2622	. . . . . . . . . . 11 |- (Sect ` C ) = (Sect ` C )
22	2, 3, 14, 15, 16, 21	isinv 16420	. . . . . . . . . 10 |- ( ( ph /\ F e. ( X I Y ) ) -> ( F ( X (Inv ` C ) Y ) ( ( X (Inv ` C ) Y ) ` F ) <-> ( F ( X (Sect ` C ) Y ) ( ( X (Inv ` C ) Y ) ` F ) /\ ( ( X (Inv ` C ) Y ) ` F ) ( Y (Sect ` C ) X ) F ) ) )
23	20, 22	mpbid 222	. . . . . . . . 9 |- ( ( ph /\ F e. ( X I Y ) ) -> ( F ( X (Sect ` C ) Y ) ( ( X (Inv ` C ) Y ) ` F ) /\ ( ( X (Inv ` C ) Y ) ` F ) ( Y (Sect ` C ) X ) F ) )
24	23	simpld 475	. . . . . . . 8 |- ( ( ph /\ F e. ( X I Y ) ) -> F ( X (Sect ` C ) Y ) ( ( X (Inv ` C ) Y ) ` F ) )
25		eqid 2622	. . . . . . . . 9 |- ( Hom ` C ) = ( Hom ` C )
26		eqid 2622	. . . . . . . . 9 |- (comp ` C ) = (comp ` C )
27		eqid 2622	. . . . . . . . 9 |- ( Id ` C ) = ( Id ` C )
28	2, 25, 26, 27, 21, 14, 15, 16	issect 16413	. . . . . . . 8 |- ( ( ph /\ F e. ( X I Y ) ) -> ( F ( X (Sect ` C ) Y ) ( ( X (Inv ` C ) Y ) ` F ) <-> ( F e. ( X ( Hom ` C ) Y ) /\ ( ( X (Inv ` C ) Y ) ` F ) e. ( Y ( Hom ` C ) X ) /\ ( ( ( X (Inv ` C ) Y ) ` F ) ( <. X , Y >. (comp ` C ) X ) F ) = ( ( Id ` C ) ` X ) ) ) )
29	24, 28	mpbid 222	. . . . . . 7 |- ( ( ph /\ F e. ( X I Y ) ) -> ( F e. ( X ( Hom ` C ) Y ) /\ ( ( X (Inv ` C ) Y ) ` F ) e. ( Y ( Hom ` C ) X ) /\ ( ( ( X (Inv ` C ) Y ) ` F ) ( <. X , Y >. (comp ` C ) X ) F ) = ( ( Id ` C ) ` X ) ) )
30	29	simp1d 1073	. . . . . 6 |- ( ( ph /\ F e. ( X I Y ) ) -> F e. ( X ( Hom ` C ) Y ) )
31	5, 2, 4, 25, 8, 9	catchom 16749	. . . . . . 7 |- ( ph -> ( X ( Hom ` C ) Y ) = ( X Func Y ) )
32	31	adantr 481	. . . . . 6 |- ( ( ph /\ F e. ( X I Y ) ) -> ( X ( Hom ` C ) Y ) = ( X Func Y ) )
33	30, 32	eleqtrd 2703	. . . . 5 |- ( ( ph /\ F e. ( X I Y ) ) -> F e. ( X Func Y ) )
34		1st2nd 7214	. . . . 5 |- ( ( Rel ( X Func Y ) /\ F e. ( X Func Y ) ) -> F = <. ( 1st ` F ) , ( 2nd ` F ) >. )
35	1, 33, 34	sylancr 695	. . . 4 |- ( ( ph /\ F e. ( X I Y ) ) -> F = <. ( 1st ` F ) , ( 2nd ` F ) >. )
36		1st2ndbr 7217	. . . . . . 7 |- ( ( Rel ( X Func Y ) /\ F e. ( X Func Y ) ) -> ( 1st ` F ) ( X Func Y ) ( 2nd ` F ) )
37	1, 33, 36	sylancr 695	. . . . . 6 |- ( ( ph /\ F e. ( X I Y ) ) -> ( 1st ` F ) ( X Func Y ) ( 2nd ` F ) )
38		catciso.r	. . . . . . . . 9 |- R = ( Base ` X )
39		eqid 2622	. . . . . . . . 9 |- ( Hom ` X ) = ( Hom ` X )
40		eqid 2622	. . . . . . . . 9 |- ( Hom ` Y ) = ( Hom ` Y )
41	37	adantr 481	. . . . . . . . 9 |- ( ( ( ph /\ F e. ( X I Y ) ) /\ ( x e. R /\ y e. R ) ) -> ( 1st ` F ) ( X Func Y ) ( 2nd ` F ) )
42		simprl 794	. . . . . . . . 9 |- ( ( ( ph /\ F e. ( X I Y ) ) /\ ( x e. R /\ y e. R ) ) -> x e. R )
43		simprr 796	. . . . . . . . 9 |- ( ( ( ph /\ F e. ( X I Y ) ) /\ ( x e. R /\ y e. R ) ) -> y e. R )
44	38, 39, 40, 41, 42, 43	funcf2 16528	. . . . . . . 8 |- ( ( ( ph /\ F e. ( X I Y ) ) /\ ( x e. R /\ y e. R ) ) -> ( x ( 2nd ` F ) y ) : ( x ( Hom ` X ) y ) --> ( ( ( 1st ` F ) ` x ) ( Hom ` Y ) ( ( 1st ` F ) ` y ) ) )
45		catciso.s	. . . . . . . . . 10 |- S = ( Base ` Y )
46		relfunc 16522	. . . . . . . . . . . 12 |- Rel ( Y Func X )
47	29	simp2d 1074	. . . . . . . . . . . . 13 |- ( ( ph /\ F e. ( X I Y ) ) -> ( ( X (Inv ` C ) Y ) ` F ) e. ( Y ( Hom ` C ) X ) )
48	5, 2, 4, 25, 9, 8	catchom 16749	. . . . . . . . . . . . . 14 |- ( ph -> ( Y ( Hom ` C ) X ) = ( Y Func X ) )
49	48	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ F e. ( X I Y ) ) -> ( Y ( Hom ` C ) X ) = ( Y Func X ) )
50	47, 49	eleqtrd 2703	. . . . . . . . . . . 12 |- ( ( ph /\ F e. ( X I Y ) ) -> ( ( X (Inv ` C ) Y ) ` F ) e. ( Y Func X ) )
51		1st2ndbr 7217	. . . . . . . . . . . 12 |- ( ( Rel ( Y Func X ) /\ ( ( X (Inv ` C ) Y ) ` F ) e. ( Y Func X ) ) -> ( 1st ` ( ( X (Inv ` C ) Y ) ` F ) ) ( Y Func X ) ( 2nd ` ( ( X (Inv ` C ) Y ) ` F ) ) )
52	46, 50, 51	sylancr 695	. . . . . . . . . . 11 |- ( ( ph /\ F e. ( X I Y ) ) -> ( 1st ` ( ( X (Inv ` C ) Y ) ` F ) ) ( Y Func X ) ( 2nd ` ( ( X (Inv ` C ) Y ) ` F ) ) )
53	52	adantr 481	. . . . . . . . . 10 |- ( ( ( ph /\ F e. ( X I Y ) ) /\ ( x e. R /\ y e. R ) ) -> ( 1st ` ( ( X (Inv ` C ) Y ) ` F ) ) ( Y Func X ) ( 2nd ` ( ( X (Inv ` C ) Y ) ` F ) ) )
54	38, 45, 41	funcf1 16526	. . . . . . . . . . 11 |- ( ( ( ph /\ F e. ( X I Y ) ) /\ ( x e. R /\ y e. R ) ) -> ( 1st ` F ) : R --> S )
55	54, 42	ffvelrnd 6360	. . . . . . . . . 10 |- ( ( ( ph /\ F e. ( X I Y ) ) /\ ( x e. R /\ y e. R ) ) -> ( ( 1st ` F ) ` x ) e. S )
56	54, 43	ffvelrnd 6360	. . . . . . . . . 10 |- ( ( ( ph /\ F e. ( X I Y ) ) /\ ( x e. R /\ y e. R ) ) -> ( ( 1st ` F ) ` y ) e. S )
57	45, 40, 39, 53, 55, 56	funcf2 16528	. . . . . . . . 9 |- ( ( ( ph /\ F e. ( X I Y ) ) /\ ( x e. R /\ y e. R ) ) -> ( ( ( 1st ` F ) ` x ) ( 2nd ` ( ( X (Inv ` C ) Y ) ` F ) ) ( ( 1st ` F ) ` y ) ) : ( ( ( 1st ` F ) ` x ) ( Hom ` Y ) ( ( 1st ` F ) ` y ) ) --> ( ( ( 1st ` ( ( X (Inv ` C ) Y ) ` F ) ) ` ( ( 1st ` F ) ` x ) ) ( Hom ` X ) ( ( 1st ` ( ( X (Inv ` C ) Y ) ` F ) ) ` ( ( 1st ` F ) ` y ) ) ) )
58	29	simp3d 1075	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ F e. ( X I Y ) ) -> ( ( ( X (Inv ` C ) Y ) ` F ) ( <. X , Y >. (comp ` C ) X ) F ) = ( ( Id ` C ) ` X ) )
59	4	adantr 481	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ F e. ( X I Y ) ) -> U e. V )
60	5, 2, 59, 26, 15, 16, 15, 33, 50	catcco 16751	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ F e. ( X I Y ) ) -> ( ( ( X (Inv ` C ) Y ) ` F ) ( <. X , Y >. (comp ` C ) X ) F ) = ( ( ( X (Inv ` C ) Y ) ` F ) o.func F ) )
61		eqid 2622	. . . . . . . . . . . . . . . . . 18 |- (idfunc ` X ) = (idfunc ` X )
62	5, 2, 27, 61, 4, 8	catcid 16753	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( ( Id ` C ) ` X ) = (idfunc ` X ) )
63	62	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ F e. ( X I Y ) ) -> ( ( Id ` C ) ` X ) = (idfunc ` X ) )
64	58, 60, 63	3eqtr3d 2664	. . . . . . . . . . . . . . 15 |- ( ( ph /\ F e. ( X I Y ) ) -> ( ( ( X (Inv ` C ) Y ) ` F ) o.func F ) = (idfunc ` X ) )
65	64	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ F e. ( X I Y ) ) /\ ( x e. R /\ y e. R ) ) -> ( ( ( X (Inv ` C ) Y ) ` F ) o.func F ) = (idfunc ` X ) )
66	65	fveq2d 6195	. . . . . . . . . . . . 13 |- ( ( ( ph /\ F e. ( X I Y ) ) /\ ( x e. R /\ y e. R ) ) -> ( 1st ` ( ( ( X (Inv ` C ) Y ) ` F ) o.func F ) ) = ( 1st ` (idfunc ` X ) ) )
67	66	fveq1d 6193	. . . . . . . . . . . 12 |- ( ( ( ph /\ F e. ( X I Y ) ) /\ ( x e. R /\ y e. R ) ) -> ( ( 1st ` ( ( ( X (Inv ` C ) Y ) ` F ) o.func F ) ) ` x ) = ( ( 1st ` (idfunc ` X ) ) ` x ) )
68	33	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( ph /\ F e. ( X I Y ) ) /\ ( x e. R /\ y e. R ) ) -> F e. ( X Func Y ) )
69	50	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( ph /\ F e. ( X I Y ) ) /\ ( x e. R /\ y e. R ) ) -> ( ( X (Inv ` C ) Y ) ` F ) e. ( Y Func X ) )
70	38, 68, 69, 42	cofu1 16544	. . . . . . . . . . . 12 |- ( ( ( ph /\ F e. ( X I Y ) ) /\ ( x e. R /\ y e. R ) ) -> ( ( 1st ` ( ( ( X (Inv ` C ) Y ) ` F ) o.func F ) ) ` x ) = ( ( 1st ` ( ( X (Inv ` C ) Y ) ` F ) ) ` ( ( 1st ` F ) ` x ) ) )
71	5, 2, 4	catcbas 16747	. . . . . . . . . . . . . . . 16 |- ( ph -> B = ( U i^i Cat ) )
72		inss2 3834	. . . . . . . . . . . . . . . 16 |- ( U i^i Cat ) C_ Cat
73	71, 72	syl6eqss 3655	. . . . . . . . . . . . . . 15 |- ( ph -> B C_ Cat )
74	73, 8	sseldd 3604	. . . . . . . . . . . . . 14 |- ( ph -> X e. Cat )
75	74	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( ph /\ F e. ( X I Y ) ) /\ ( x e. R /\ y e. R ) ) -> X e. Cat )
76	61, 38, 75, 42	idfu1 16540	. . . . . . . . . . . 12 |- ( ( ( ph /\ F e. ( X I Y ) ) /\ ( x e. R /\ y e. R ) ) -> ( ( 1st ` (idfunc ` X ) ) ` x ) = x )
77	67, 70, 76	3eqtr3d 2664	. . . . . . . . . . 11 |- ( ( ( ph /\ F e. ( X I Y ) ) /\ ( x e. R /\ y e. R ) ) -> ( ( 1st ` ( ( X (Inv ` C ) Y ) ` F ) ) ` ( ( 1st ` F ) ` x ) ) = x )
78	66	fveq1d 6193	. . . . . . . . . . . 12 |- ( ( ( ph /\ F e. ( X I Y ) ) /\ ( x e. R /\ y e. R ) ) -> ( ( 1st ` ( ( ( X (Inv ` C ) Y ) ` F ) o.func F ) ) ` y ) = ( ( 1st ` (idfunc ` X ) ) ` y ) )
79	38, 68, 69, 43	cofu1 16544	. . . . . . . . . . . 12 |- ( ( ( ph /\ F e. ( X I Y ) ) /\ ( x e. R /\ y e. R ) ) -> ( ( 1st ` ( ( ( X (Inv ` C ) Y ) ` F ) o.func F ) ) ` y ) = ( ( 1st ` ( ( X (Inv ` C ) Y ) ` F ) ) ` ( ( 1st ` F ) ` y ) ) )
80	61, 38, 75, 43	idfu1 16540	. . . . . . . . . . . 12 |- ( ( ( ph /\ F e. ( X I Y ) ) /\ ( x e. R /\ y e. R ) ) -> ( ( 1st ` (idfunc ` X ) ) ` y ) = y )
81	78, 79, 80	3eqtr3d 2664	. . . . . . . . . . 11 |- ( ( ( ph /\ F e. ( X I Y ) ) /\ ( x e. R /\ y e. R ) ) -> ( ( 1st ` ( ( X (Inv ` C ) Y ) ` F ) ) ` ( ( 1st ` F ) ` y ) ) = y )
82	77, 81	oveq12d 6668	. . . . . . . . . 10 |- ( ( ( ph /\ F e. ( X I Y ) ) /\ ( x e. R /\ y e. R ) ) -> ( ( ( 1st ` ( ( X (Inv ` C ) Y ) ` F ) ) ` ( ( 1st ` F ) ` x ) ) ( Hom ` X ) ( ( 1st ` ( ( X (Inv ` C ) Y ) ` F ) ) ` ( ( 1st ` F ) ` y ) ) ) = ( x ( Hom ` X ) y ) )
83	82	feq3d 6032	. . . . . . . . 9 |- ( ( ( ph /\ F e. ( X I Y ) ) /\ ( x e. R /\ y e. R ) ) -> ( ( ( ( 1st ` F ) ` x ) ( 2nd ` ( ( X (Inv ` C ) Y ) ` F ) ) ( ( 1st ` F ) ` y ) ) : ( ( ( 1st ` F ) ` x ) ( Hom ` Y ) ( ( 1st ` F ) ` y ) ) --> ( ( ( 1st ` ( ( X (Inv ` C ) Y ) ` F ) ) ` ( ( 1st ` F ) ` x ) ) ( Hom ` X ) ( ( 1st ` ( ( X (Inv ` C ) Y ) ` F ) ) ` ( ( 1st ` F ) ` y ) ) ) <-> ( ( ( 1st ` F ) ` x ) ( 2nd ` ( ( X (Inv ` C ) Y ) ` F ) ) ( ( 1st ` F ) ` y ) ) : ( ( ( 1st ` F ) ` x ) ( Hom ` Y ) ( ( 1st ` F ) ` y ) ) --> ( x ( Hom ` X ) y ) ) )
84	57, 83	mpbid 222	. . . . . . . 8 |- ( ( ( ph /\ F e. ( X I Y ) ) /\ ( x e. R /\ y e. R ) ) -> ( ( ( 1st ` F ) ` x ) ( 2nd ` ( ( X (Inv ` C ) Y ) ` F ) ) ( ( 1st ` F ) ` y ) ) : ( ( ( 1st ` F ) ` x ) ( Hom ` Y ) ( ( 1st ` F ) ` y ) ) --> ( x ( Hom ` X ) y ) )
85	65	fveq2d 6195	. . . . . . . . . 10 |- ( ( ( ph /\ F e. ( X I Y ) ) /\ ( x e. R /\ y e. R ) ) -> ( 2nd ` ( ( ( X (Inv ` C ) Y ) ` F ) o.func F ) ) = ( 2nd ` (idfunc ` X ) ) )
86	85	oveqd 6667	. . . . . . . . 9 |- ( ( ( ph /\ F e. ( X I Y ) ) /\ ( x e. R /\ y e. R ) ) -> ( x ( 2nd ` ( ( ( X (Inv ` C ) Y ) ` F ) o.func F ) ) y ) = ( x ( 2nd ` (idfunc ` X ) ) y ) )
87	38, 68, 69, 42, 43	cofu2nd 16545	. . . . . . . . 9 |- ( ( ( ph /\ F e. ( X I Y ) ) /\ ( x e. R /\ y e. R ) ) -> ( x ( 2nd ` ( ( ( X (Inv ` C ) Y ) ` F ) o.func F ) ) y ) = ( ( ( ( 1st ` F ) ` x ) ( 2nd ` ( ( X (Inv ` C ) Y ) ` F ) ) ( ( 1st ` F ) ` y ) ) o. ( x ( 2nd ` F ) y ) ) )
88	61, 38, 75, 39, 42, 43	idfu2nd 16537	. . . . . . . . 9 |- ( ( ( ph /\ F e. ( X I Y ) ) /\ ( x e. R /\ y e. R ) ) -> ( x ( 2nd ` (idfunc ` X ) ) y ) = ( _I |` ( x ( Hom ` X ) y ) ) )
89	86, 87, 88	3eqtr3d 2664	. . . . . . . 8 |- ( ( ( ph /\ F e. ( X I Y ) ) /\ ( x e. R /\ y e. R ) ) -> ( ( ( ( 1st ` F ) ` x ) ( 2nd ` ( ( X (Inv ` C ) Y ) ` F ) ) ( ( 1st ` F ) ` y ) ) o. ( x ( 2nd ` F ) y ) ) = ( _I |` ( x ( Hom ` X ) y ) ) )
90	23	simprd 479	. . . . . . . . . . . . . . 15 |- ( ( ph /\ F e. ( X I Y ) ) -> ( ( X (Inv ` C ) Y ) ` F ) ( Y (Sect ` C ) X ) F )
91	2, 25, 26, 27, 21, 14, 16, 15	issect 16413	. . . . . . . . . . . . . . 15 |- ( ( ph /\ F e. ( X I Y ) ) -> ( ( ( X (Inv ` C ) Y ) ` F ) ( Y (Sect ` C ) X ) F <-> ( ( ( X (Inv ` C ) Y ) ` F ) e. ( Y ( Hom ` C ) X ) /\ F e. ( X ( Hom ` C ) Y ) /\ ( F ( <. Y , X >. (comp ` C ) Y ) ( ( X (Inv ` C ) Y ) ` F ) ) = ( ( Id ` C ) ` Y ) ) ) )
92	90, 91	mpbid 222	. . . . . . . . . . . . . 14 |- ( ( ph /\ F e. ( X I Y ) ) -> ( ( ( X (Inv ` C ) Y ) ` F ) e. ( Y ( Hom ` C ) X ) /\ F e. ( X ( Hom ` C ) Y ) /\ ( F ( <. Y , X >. (comp ` C ) Y ) ( ( X (Inv ` C ) Y ) ` F ) ) = ( ( Id ` C ) ` Y ) ) )
93	92	simp3d 1075	. . . . . . . . . . . . 13 |- ( ( ph /\ F e. ( X I Y ) ) -> ( F ( <. Y , X >. (comp ` C ) Y ) ( ( X (Inv ` C ) Y ) ` F ) ) = ( ( Id ` C ) ` Y ) )
94	5, 2, 59, 26, 16, 15, 16, 50, 33	catcco 16751	. . . . . . . . . . . . 13 |- ( ( ph /\ F e. ( X I Y ) ) -> ( F ( <. Y , X >. (comp ` C ) Y ) ( ( X (Inv ` C ) Y ) ` F ) ) = ( F o.func ( ( X (Inv ` C ) Y ) ` F ) ) )
95		eqid 2622	. . . . . . . . . . . . . . 15 |- (idfunc ` Y ) = (idfunc ` Y )
96	5, 2, 27, 95, 4, 9	catcid 16753	. . . . . . . . . . . . . 14 |- ( ph -> ( ( Id ` C ) ` Y ) = (idfunc ` Y ) )
97	96	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ F e. ( X I Y ) ) -> ( ( Id ` C ) ` Y ) = (idfunc ` Y ) )
98	93, 94, 97	3eqtr3d 2664	. . . . . . . . . . . 12 |- ( ( ph /\ F e. ( X I Y ) ) -> ( F o.func ( ( X (Inv ` C ) Y ) ` F ) ) = (idfunc ` Y ) )
99	98	adantr 481	. . . . . . . . . . 11 |- ( ( ( ph /\ F e. ( X I Y ) ) /\ ( x e. R /\ y e. R ) ) -> ( F o.func ( ( X (Inv ` C ) Y ) ` F ) ) = (idfunc ` Y ) )
100	99	fveq2d 6195	. . . . . . . . . 10 |- ( ( ( ph /\ F e. ( X I Y ) ) /\ ( x e. R /\ y e. R ) ) -> ( 2nd ` ( F o.func ( ( X (Inv ` C ) Y ) ` F ) ) ) = ( 2nd ` (idfunc ` Y ) ) )
101	100	oveqd 6667	. . . . . . . . 9 |- ( ( ( ph /\ F e. ( X I Y ) ) /\ ( x e. R /\ y e. R ) ) -> ( ( ( 1st ` F ) ` x ) ( 2nd ` ( F o.func ( ( X (Inv ` C ) Y ) ` F ) ) ) ( ( 1st ` F ) ` y ) ) = ( ( ( 1st ` F ) ` x ) ( 2nd ` (idfunc ` Y ) ) ( ( 1st ` F ) ` y ) ) )
102	45, 69, 68, 55, 56	cofu2nd 16545	. . . . . . . . . 10 |- ( ( ( ph /\ F e. ( X I Y ) ) /\ ( x e. R /\ y e. R ) ) -> ( ( ( 1st ` F ) ` x ) ( 2nd ` ( F o.func ( ( X (Inv ` C ) Y ) ` F ) ) ) ( ( 1st ` F ) ` y ) ) = ( ( ( ( 1st ` ( ( X (Inv ` C ) Y ) ` F ) ) ` ( ( 1st ` F ) ` x ) ) ( 2nd ` F ) ( ( 1st ` ( ( X (Inv ` C ) Y ) ` F ) ) ` ( ( 1st ` F ) ` y ) ) ) o. ( ( ( 1st ` F ) ` x ) ( 2nd ` ( ( X (Inv ` C ) Y ) ` F ) ) ( ( 1st ` F ) ` y ) ) ) )
103	77, 81	oveq12d 6668	. . . . . . . . . . 11 |- ( ( ( ph /\ F e. ( X I Y ) ) /\ ( x e. R /\ y e. R ) ) -> ( ( ( 1st ` ( ( X (Inv ` C ) Y ) ` F ) ) ` ( ( 1st ` F ) ` x ) ) ( 2nd ` F ) ( ( 1st ` ( ( X (Inv ` C ) Y ) ` F ) ) ` ( ( 1st ` F ) ` y ) ) ) = ( x ( 2nd ` F ) y ) )
104	103	coeq1d 5283	. . . . . . . . . 10 |- ( ( ( ph /\ F e. ( X I Y ) ) /\ ( x e. R /\ y e. R ) ) -> ( ( ( ( 1st ` ( ( X (Inv ` C ) Y ) ` F ) ) ` ( ( 1st ` F ) ` x ) ) ( 2nd ` F ) ( ( 1st ` ( ( X (Inv ` C ) Y ) ` F ) ) ` ( ( 1st ` F ) ` y ) ) ) o. ( ( ( 1st ` F ) ` x ) ( 2nd ` ( ( X (Inv ` C ) Y ) ` F ) ) ( ( 1st ` F ) ` y ) ) ) = ( ( x ( 2nd ` F ) y ) o. ( ( ( 1st ` F ) ` x ) ( 2nd ` ( ( X (Inv ` C ) Y ) ` F ) ) ( ( 1st ` F ) ` y ) ) ) )
105	102, 104	eqtrd 2656	. . . . . . . . 9 |- ( ( ( ph /\ F e. ( X I Y ) ) /\ ( x e. R /\ y e. R ) ) -> ( ( ( 1st ` F ) ` x ) ( 2nd ` ( F o.func ( ( X (Inv ` C ) Y ) ` F ) ) ) ( ( 1st ` F ) ` y ) ) = ( ( x ( 2nd ` F ) y ) o. ( ( ( 1st ` F ) ` x ) ( 2nd ` ( ( X (Inv ` C ) Y ) ` F ) ) ( ( 1st ` F ) ` y ) ) ) )
106	73	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ F e. ( X I Y ) ) /\ ( x e. R /\ y e. R ) ) -> B C_ Cat )
107	9	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ F e. ( X I Y ) ) /\ ( x e. R /\ y e. R ) ) -> Y e. B )
108	106, 107	sseldd 3604	. . . . . . . . . 10 |- ( ( ( ph /\ F e. ( X I Y ) ) /\ ( x e. R /\ y e. R ) ) -> Y e. Cat )
109	95, 45, 108, 40, 55, 56	idfu2nd 16537	. . . . . . . . 9 |- ( ( ( ph /\ F e. ( X I Y ) ) /\ ( x e. R /\ y e. R ) ) -> ( ( ( 1st ` F ) ` x ) ( 2nd ` (idfunc ` Y ) ) ( ( 1st ` F ) ` y ) ) = ( _I |` ( ( ( 1st ` F ) ` x ) ( Hom ` Y ) ( ( 1st ` F ) ` y ) ) ) )
110	101, 105, 109	3eqtr3d 2664	. . . . . . . 8 |- ( ( ( ph /\ F e. ( X I Y ) ) /\ ( x e. R /\ y e. R ) ) -> ( ( x ( 2nd ` F ) y ) o. ( ( ( 1st ` F ) ` x ) ( 2nd ` ( ( X (Inv ` C ) Y ) ` F ) ) ( ( 1st ` F ) ` y ) ) ) = ( _I |` ( ( ( 1st ` F ) ` x ) ( Hom ` Y ) ( ( 1st ` F ) ` y ) ) ) )
111	44, 84, 89, 110	fcof1od 6549	. . . . . . 7 |- ( ( ( ph /\ F e. ( X I Y ) ) /\ ( x e. R /\ y e. R ) ) -> ( x ( 2nd ` F ) y ) : ( x ( Hom ` X ) y ) -1-1-onto-> ( ( ( 1st ` F ) ` x ) ( Hom ` Y ) ( ( 1st ` F ) ` y ) ) )
112	111	ralrimivva 2971	. . . . . 6 |- ( ( ph /\ F e. ( X I Y ) ) -> A. x e. R A. y e. R ( x ( 2nd ` F ) y ) : ( x ( Hom ` X ) y ) -1-1-onto-> ( ( ( 1st ` F ) ` x ) ( Hom ` Y ) ( ( 1st ` F ) ` y ) ) )
113	38, 39, 40	isffth2 16576	. . . . . 6 |- ( ( 1st ` F ) ( ( X Full Y ) i^i ( X Faith Y ) ) ( 2nd ` F ) <-> ( ( 1st ` F ) ( X Func Y ) ( 2nd ` F ) /\ A. x e. R A. y e. R ( x ( 2nd ` F ) y ) : ( x ( Hom ` X ) y ) -1-1-onto-> ( ( ( 1st ` F ) ` x ) ( Hom ` Y ) ( ( 1st ` F ) ` y ) ) ) )
114	37, 112, 113	sylanbrc 698	. . . . 5 |- ( ( ph /\ F e. ( X I Y ) ) -> ( 1st ` F ) ( ( X Full Y ) i^i ( X Faith Y ) ) ( 2nd ` F ) )
115		df-br 4654	. . . . 5 |- ( ( 1st ` F ) ( ( X Full Y ) i^i ( X Faith Y ) ) ( 2nd ` F ) <-> <. ( 1st ` F ) , ( 2nd ` F ) >. e. ( ( X Full Y ) i^i ( X Faith Y ) ) )
116	114, 115	sylib 208	. . . 4 |- ( ( ph /\ F e. ( X I Y ) ) -> <. ( 1st ` F ) , ( 2nd ` F ) >. e. ( ( X Full Y ) i^i ( X Faith Y ) ) )
117	35, 116	eqeltrd 2701	. . 3 |- ( ( ph /\ F e. ( X I Y ) ) -> F e. ( ( X Full Y ) i^i ( X Faith Y ) ) )
118	38, 45, 37	funcf1 16526	. . . 4 |- ( ( ph /\ F e. ( X I Y ) ) -> ( 1st ` F ) : R --> S )
119	45, 38, 52	funcf1 16526	. . . 4 |- ( ( ph /\ F e. ( X I Y ) ) -> ( 1st ` ( ( X (Inv ` C ) Y ) ` F ) ) : S --> R )
120	64	fveq2d 6195	. . . . 5 |- ( ( ph /\ F e. ( X I Y ) ) -> ( 1st ` ( ( ( X (Inv ` C ) Y ) ` F ) o.func F ) ) = ( 1st ` (idfunc ` X ) ) )
121	38, 33, 50	cofu1st 16543	. . . . 5 |- ( ( ph /\ F e. ( X I Y ) ) -> ( 1st ` ( ( ( X (Inv ` C ) Y ) ` F ) o.func F ) ) = ( ( 1st ` ( ( X (Inv ` C ) Y ) ` F ) ) o. ( 1st ` F ) ) )
122	74	adantr 481	. . . . . 6 |- ( ( ph /\ F e. ( X I Y ) ) -> X e. Cat )
123	61, 38, 122	idfu1st 16539	. . . . 5 |- ( ( ph /\ F e. ( X I Y ) ) -> ( 1st ` (idfunc ` X ) ) = ( _I |` R ) )
124	120, 121, 123	3eqtr3d 2664	. . . 4 |- ( ( ph /\ F e. ( X I Y ) ) -> ( ( 1st ` ( ( X (Inv ` C ) Y ) ` F ) ) o. ( 1st ` F ) ) = ( _I |` R ) )
125	98	fveq2d 6195	. . . . 5 |- ( ( ph /\ F e. ( X I Y ) ) -> ( 1st ` ( F o.func ( ( X (Inv ` C ) Y ) ` F ) ) ) = ( 1st ` (idfunc ` Y ) ) )
126	45, 50, 33	cofu1st 16543	. . . . 5 |- ( ( ph /\ F e. ( X I Y ) ) -> ( 1st ` ( F o.func ( ( X (Inv ` C ) Y ) ` F ) ) ) = ( ( 1st ` F ) o. ( 1st ` ( ( X (Inv ` C ) Y ) ` F ) ) ) )
127	73, 9	sseldd 3604	. . . . . . 7 |- ( ph -> Y e. Cat )
128	127	adantr 481	. . . . . 6 |- ( ( ph /\ F e. ( X I Y ) ) -> Y e. Cat )
129	95, 45, 128	idfu1st 16539	. . . . 5 |- ( ( ph /\ F e. ( X I Y ) ) -> ( 1st ` (idfunc ` Y ) ) = ( _I |` S ) )
130	125, 126, 129	3eqtr3d 2664	. . . 4 |- ( ( ph /\ F e. ( X I Y ) ) -> ( ( 1st ` F ) o. ( 1st ` ( ( X (Inv ` C ) Y ) ` F ) ) ) = ( _I |` S ) )
131	118, 119, 124, 130	fcof1od 6549	. . 3 |- ( ( ph /\ F e. ( X I Y ) ) -> ( 1st ` F ) : R -1-1-onto-> S )
132	117, 131	jca 554	. 2 |- ( ( ph /\ F e. ( X I Y ) ) -> ( F e. ( ( X Full Y ) i^i ( X Faith Y ) ) /\ ( 1st ` F ) : R -1-1-onto-> S ) )
133	7	adantr 481	. . 3 |- ( ( ph /\ ( F e. ( ( X Full Y ) i^i ( X Faith Y ) ) /\ ( 1st ` F ) : R -1-1-onto-> S ) ) -> C e. Cat )
134	8	adantr 481	. . 3 |- ( ( ph /\ ( F e. ( ( X Full Y ) i^i ( X Faith Y ) ) /\ ( 1st ` F ) : R -1-1-onto-> S ) ) -> X e. B )
135	9	adantr 481	. . 3 |- ( ( ph /\ ( F e. ( ( X Full Y ) i^i ( X Faith Y ) ) /\ ( 1st ` F ) : R -1-1-onto-> S ) ) -> Y e. B )
136		inss1 3833	. . . . . . 7 |- ( ( X Full Y ) i^i ( X Faith Y ) ) C_ ( X Full Y )
137		fullfunc 16566	. . . . . . 7 |- ( X Full Y ) C_ ( X Func Y )
138	136, 137	sstri 3612	. . . . . 6 |- ( ( X Full Y ) i^i ( X Faith Y ) ) C_ ( X Func Y )
139		simprl 794	. . . . . 6 |- ( ( ph /\ ( F e. ( ( X Full Y ) i^i ( X Faith Y ) ) /\ ( 1st ` F ) : R -1-1-onto-> S ) ) -> F e. ( ( X Full Y ) i^i ( X Faith Y ) ) )
140	138, 139	sseldi 3601	. . . . 5 |- ( ( ph /\ ( F e. ( ( X Full Y ) i^i ( X Faith Y ) ) /\ ( 1st ` F ) : R -1-1-onto-> S ) ) -> F e. ( X Func Y ) )
141	1, 140, 34	sylancr 695	. . . 4 |- ( ( ph /\ ( F e. ( ( X Full Y ) i^i ( X Faith Y ) ) /\ ( 1st ` F ) : R -1-1-onto-> S ) ) -> F = <. ( 1st ` F ) , ( 2nd ` F ) >. )
142	4	adantr 481	. . . . 5 |- ( ( ph /\ ( F e. ( ( X Full Y ) i^i ( X Faith Y ) ) /\ ( 1st ` F ) : R -1-1-onto-> S ) ) -> U e. V )
143		eqid 2622	. . . . 5 |- ( x e. S , y e. S |-> `' ( ( `' ( 1st ` F ) ` x ) ( 2nd ` F ) ( `' ( 1st ` F ) ` y ) ) ) = ( x e. S , y e. S |-> `' ( ( `' ( 1st ` F ) ` x ) ( 2nd ` F ) ( `' ( 1st ` F ) ` y ) ) )
144	141, 139	eqeltrrd 2702	. . . . . 6 |- ( ( ph /\ ( F e. ( ( X Full Y ) i^i ( X Faith Y ) ) /\ ( 1st ` F ) : R -1-1-onto-> S ) ) -> <. ( 1st ` F ) , ( 2nd ` F ) >. e. ( ( X Full Y ) i^i ( X Faith Y ) ) )
145	144, 115	sylibr 224	. . . . 5 |- ( ( ph /\ ( F e. ( ( X Full Y ) i^i ( X Faith Y ) ) /\ ( 1st ` F ) : R -1-1-onto-> S ) ) -> ( 1st ` F ) ( ( X Full Y ) i^i ( X Faith Y ) ) ( 2nd ` F ) )
146		simprr 796	. . . . 5 |- ( ( ph /\ ( F e. ( ( X Full Y ) i^i ( X Faith Y ) ) /\ ( 1st ` F ) : R -1-1-onto-> S ) ) -> ( 1st ` F ) : R -1-1-onto-> S )
147	5, 2, 38, 45, 142, 134, 135, 3, 143, 145, 146	catcisolem 16756	. . . 4 |- ( ( ph /\ ( F e. ( ( X Full Y ) i^i ( X Faith Y ) ) /\ ( 1st ` F ) : R -1-1-onto-> S ) ) -> <. ( 1st ` F ) , ( 2nd ` F ) >. ( X (Inv ` C ) Y ) <. `' ( 1st ` F ) , ( x e. S , y e. S |-> `' ( ( `' ( 1st ` F ) ` x ) ( 2nd ` F ) ( `' ( 1st ` F ) ` y ) ) ) >. )
148	141, 147	eqbrtrd 4675	. . 3 |- ( ( ph /\ ( F e. ( ( X Full Y ) i^i ( X Faith Y ) ) /\ ( 1st ` F ) : R -1-1-onto-> S ) ) -> F ( X (Inv ` C ) Y ) <. `' ( 1st ` F ) , ( x e. S , y e. S |-> `' ( ( `' ( 1st ` F ) ` x ) ( 2nd ` F ) ( `' ( 1st ` F ) ` y ) ) ) >. )
149	2, 3, 133, 134, 135, 10, 148	inviso1 16426	. 2 |- ( ( ph /\ ( F e. ( ( X Full Y ) i^i ( X Faith Y ) ) /\ ( 1st ` F ) : R -1-1-onto-> S ) ) -> F e. ( X I Y ) )
150	132, 149	impbida 877	1 |- ( ph -> ( F e. ( X I Y ) <-> ( F e. ( ( X Full Y ) i^i ( X Faith Y ) ) /\ ( 1st ` F ) : R -1-1-onto-> S ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   i^i cin 3573   C_ wss 3574   <.cop 4183   class class class wbr 4653   _I cid 5023   `'ccnv 5113   dom cdm 5114   |` cres 5116   o. ccom 5118   Rel wrel 5119   Fun wfun 5882   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   Basecbs 15857   Hom chom 15952  compcco 15953   Catccat 16325   Idccid 16326  Sectcsect 16404  Invcinv 16405   Iso ciso 16406   Func cfunc 16514  id_funccidfu 16515   o._func ccofu 16516   Full cful 16562   Faith cfth 16563  CatCatccatc 16744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-hom 15966  df-cco 15967  df-cat 16329  df-cid 16330  df-sect 16407  df-inv 16408  df-iso 16409  df-func 16518  df-idfu 16519  df-cofu 16520  df-full 16564  df-fth 16565  df-catc 16745

This theorem is referenced by:  yoniso  16925