Theorem yonedalem4c 16917

Description: Lemma for yoneda 16923. (Contributed by Mario Carneiro, 29-Jan-2017.)

Hypotheses

Ref	Expression
yoneda.y	|- Y = (Yon ` C )
yoneda.b	|- B = ( Base ` C )
yoneda.1	|- .1. = ( Id ` C )
yoneda.o	|- O = (oppCat ` C )
yoneda.s	|- S = ( SetCat ` U )
yoneda.t	|- T = ( SetCat ` V )
yoneda.q	|- Q = ( O FuncCat S )
yoneda.h	|- H = (HomF ` Q )
yoneda.r	|- R = ( ( Q X.c O ) FuncCat T )
yoneda.e	|- E = ( O evalF S )
yoneda.z	|- Z = ( H o.func ( ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ⟨,⟩F ( Q 1stF O ) ) )
yoneda.c	|- ( ph -> C e. Cat )
yoneda.w	|- ( ph -> V e. W )
yoneda.u	|- ( ph -> ran ( Hom f ` C ) C_ U )
yoneda.v	|- ( ph -> ( ran ( Hom f ` Q ) u. U ) C_ V )
yonedalem21.f	|- ( ph -> F e. ( O Func S ) )
yonedalem21.x	|- ( ph -> X e. B )
yonedalem4.n	|- N = ( f e. ( O Func S ) , x e. B |-> ( u e. ( ( 1st ` f ) ` x ) |-> ( y e. B |-> ( g e. ( y ( Hom ` C ) x ) |-> ( ( ( x ( 2nd ` f ) y ) ` g ) ` u ) ) ) ) )
yonedalem4.p	|- ( ph -> A e. ( ( 1st ` F ) ` X ) )

Assertion

Ref	Expression
yonedalem4c	|- ( ph -> ( ( F N X ) ` A ) e. ( ( ( 1st ` Y ) ` X ) ( O Nat S ) F ) )

Distinct variable groups:   f, g, x, y, .1.   u, g, A, y   u, f, C, g, x, y   f, E, g, u, y   f, F, g, u, x, y   B, f, g, u, x, y   f, O, g, u, x, y   S, f, g, u, x, y   Q, f, g, u, x   T, f, g, u, y   ph, f, g, u, x, y   u, R   f, Y, g, u, x, y   f, Z, g, u, x, y   f, X, g, u, x, y

Allowed substitution hints:   A( x, f)   Q( y)   R( x, y, f, g)   T( x)   U( x, y, u, f, g)   .1. ( u)   E( x)   H( x, y, u, f, g)   N( x, y, u, f, g)   V( x, y, u, f, g)   W( x, y, u, f, g)

Proof of Theorem yonedalem4c

Dummy variables h k w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		yoneda.y	. . . . 5 |- Y = (Yon ` C )
2		yoneda.b	. . . . 5 |- B = ( Base ` C )
3		yoneda.1	. . . . 5 |- .1. = ( Id ` C )
4		yoneda.o	. . . . 5 |- O = (oppCat ` C )
5		yoneda.s	. . . . 5 |- S = ( SetCat ` U )
6		yoneda.t	. . . . 5 |- T = ( SetCat ` V )
7		yoneda.q	. . . . 5 |- Q = ( O FuncCat S )
8		yoneda.h	. . . . 5 |- H = (HomF ` Q )
9		yoneda.r	. . . . 5 |- R = ( ( Q X.c O ) FuncCat T )
10		yoneda.e	. . . . 5 |- E = ( O evalF S )
11		yoneda.z	. . . . 5 |- Z = ( H o.func ( ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ⟨,⟩F ( Q 1stF O ) ) )
12		yoneda.c	. . . . 5 |- ( ph -> C e. Cat )
13		yoneda.w	. . . . 5 |- ( ph -> V e. W )
14		yoneda.u	. . . . 5 |- ( ph -> ran ( Hom f ` C ) C_ U )
15		yoneda.v	. . . . 5 |- ( ph -> ( ran ( Hom f ` Q ) u. U ) C_ V )
16		yonedalem21.f	. . . . 5 |- ( ph -> F e. ( O Func S ) )
17		yonedalem21.x	. . . . 5 |- ( ph -> X e. B )
18		yonedalem4.n	. . . . 5 |- N = ( f e. ( O Func S ) , x e. B |-> ( u e. ( ( 1st ` f ) ` x ) |-> ( y e. B |-> ( g e. ( y ( Hom ` C ) x ) |-> ( ( ( x ( 2nd ` f ) y ) ` g ) ` u ) ) ) ) )
19		yonedalem4.p	. . . . 5 |- ( ph -> A e. ( ( 1st ` F ) ` X ) )
20	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19	yonedalem4a 16915	. . . 4 |- ( ph -> ( ( F N X ) ` A ) = ( y e. B |-> ( g e. ( y ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) y ) ` g ) ` A ) ) ) )
21		oveq1 6657	. . . . . 6 |- ( y = z -> ( y ( Hom ` C ) X ) = ( z ( Hom ` C ) X ) )
22		oveq2 6658	. . . . . . . 8 |- ( y = z -> ( X ( 2nd ` F ) y ) = ( X ( 2nd ` F ) z ) )
23	22	fveq1d 6193	. . . . . . 7 |- ( y = z -> ( ( X ( 2nd ` F ) y ) ` g ) = ( ( X ( 2nd ` F ) z ) ` g ) )
24	23	fveq1d 6193	. . . . . 6 |- ( y = z -> ( ( ( X ( 2nd ` F ) y ) ` g ) ` A ) = ( ( ( X ( 2nd ` F ) z ) ` g ) ` A ) )
25	21, 24	mpteq12dv 4733	. . . . 5 |- ( y = z -> ( g e. ( y ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) y ) ` g ) ` A ) ) = ( g e. ( z ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) z ) ` g ) ` A ) ) )
26	25	cbvmptv 4750	. . . 4 |- ( y e. B |-> ( g e. ( y ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) y ) ` g ) ` A ) ) ) = ( z e. B |-> ( g e. ( z ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) z ) ` g ) ` A ) ) )
27	20, 26	syl6eq 2672	. . 3 |- ( ph -> ( ( F N X ) ` A ) = ( z e. B |-> ( g e. ( z ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) z ) ` g ) ` A ) ) ) )
28	4, 2	oppcbas 16378	. . . . . . . . . . . . 13 |- B = ( Base ` O )
29		eqid 2622	. . . . . . . . . . . . 13 |- ( Hom ` O ) = ( Hom ` O )
30		eqid 2622	. . . . . . . . . . . . 13 |- ( Hom ` S ) = ( Hom ` S )
31		relfunc 16522	. . . . . . . . . . . . . . 15 |- Rel ( O Func S )
32		1st2ndbr 7217	. . . . . . . . . . . . . . 15 |- ( ( Rel ( O Func S ) /\ F e. ( O Func S ) ) -> ( 1st ` F ) ( O Func S ) ( 2nd ` F ) )
33	31, 16, 32	sylancr 695	. . . . . . . . . . . . . 14 |- ( ph -> ( 1st ` F ) ( O Func S ) ( 2nd ` F ) )
34	33	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. B ) -> ( 1st ` F ) ( O Func S ) ( 2nd ` F ) )
35	17	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. B ) -> X e. B )
36		simpr 477	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. B ) -> z e. B )
37	28, 29, 30, 34, 35, 36	funcf2 16528	. . . . . . . . . . . 12 |- ( ( ph /\ z e. B ) -> ( X ( 2nd ` F ) z ) : ( X ( Hom ` O ) z ) --> ( ( ( 1st ` F ) ` X ) ( Hom ` S ) ( ( 1st ` F ) ` z ) ) )
38	37	adantr 481	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. B ) /\ g e. ( z ( Hom ` C ) X ) ) -> ( X ( 2nd ` F ) z ) : ( X ( Hom ` O ) z ) --> ( ( ( 1st ` F ) ` X ) ( Hom ` S ) ( ( 1st ` F ) ` z ) ) )
39		simpr 477	. . . . . . . . . . . 12 |- ( ( ( ph /\ z e. B ) /\ g e. ( z ( Hom ` C ) X ) ) -> g e. ( z ( Hom ` C ) X ) )
40		eqid 2622	. . . . . . . . . . . . 13 |- ( Hom ` C ) = ( Hom ` C )
41	40, 4	oppchom 16375	. . . . . . . . . . . 12 |- ( X ( Hom ` O ) z ) = ( z ( Hom ` C ) X )
42	39, 41	syl6eleqr 2712	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. B ) /\ g e. ( z ( Hom ` C ) X ) ) -> g e. ( X ( Hom ` O ) z ) )
43	38, 42	ffvelrnd 6360	. . . . . . . . . 10 |- ( ( ( ph /\ z e. B ) /\ g e. ( z ( Hom ` C ) X ) ) -> ( ( X ( 2nd ` F ) z ) ` g ) e. ( ( ( 1st ` F ) ` X ) ( Hom ` S ) ( ( 1st ` F ) ` z ) ) )
44	15	unssbd 3791	. . . . . . . . . . . . . 14 |- ( ph -> U C_ V )
45	13, 44	ssexd 4805	. . . . . . . . . . . . 13 |- ( ph -> U e. _V )
46	45	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ z e. B ) -> U e. _V )
47	46	adantr 481	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. B ) /\ g e. ( z ( Hom ` C ) X ) ) -> U e. _V )
48		eqid 2622	. . . . . . . . . . . . . . 15 |- ( Base ` S ) = ( Base ` S )
49	28, 48, 33	funcf1 16526	. . . . . . . . . . . . . 14 |- ( ph -> ( 1st ` F ) : B --> ( Base ` S ) )
50	5, 45	setcbas 16728	. . . . . . . . . . . . . . 15 |- ( ph -> U = ( Base ` S ) )
51	50	feq3d 6032	. . . . . . . . . . . . . 14 |- ( ph -> ( ( 1st ` F ) : B --> U <-> ( 1st ` F ) : B --> ( Base ` S ) ) )
52	49, 51	mpbird 247	. . . . . . . . . . . . 13 |- ( ph -> ( 1st ` F ) : B --> U )
53	52, 17	ffvelrnd 6360	. . . . . . . . . . . 12 |- ( ph -> ( ( 1st ` F ) ` X ) e. U )
54	53	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. B ) /\ g e. ( z ( Hom ` C ) X ) ) -> ( ( 1st ` F ) ` X ) e. U )
55	52	ffvelrnda 6359	. . . . . . . . . . . 12 |- ( ( ph /\ z e. B ) -> ( ( 1st ` F ) ` z ) e. U )
56	55	adantr 481	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. B ) /\ g e. ( z ( Hom ` C ) X ) ) -> ( ( 1st ` F ) ` z ) e. U )
57	5, 47, 30, 54, 56	elsetchom 16731	. . . . . . . . . 10 |- ( ( ( ph /\ z e. B ) /\ g e. ( z ( Hom ` C ) X ) ) -> ( ( ( X ( 2nd ` F ) z ) ` g ) e. ( ( ( 1st ` F ) ` X ) ( Hom ` S ) ( ( 1st ` F ) ` z ) ) <-> ( ( X ( 2nd ` F ) z ) ` g ) : ( ( 1st ` F ) ` X ) --> ( ( 1st ` F ) ` z ) ) )
58	43, 57	mpbid 222	. . . . . . . . 9 |- ( ( ( ph /\ z e. B ) /\ g e. ( z ( Hom ` C ) X ) ) -> ( ( X ( 2nd ` F ) z ) ` g ) : ( ( 1st ` F ) ` X ) --> ( ( 1st ` F ) ` z ) )
59	19	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ z e. B ) /\ g e. ( z ( Hom ` C ) X ) ) -> A e. ( ( 1st ` F ) ` X ) )
60	58, 59	ffvelrnd 6360	. . . . . . . 8 |- ( ( ( ph /\ z e. B ) /\ g e. ( z ( Hom ` C ) X ) ) -> ( ( ( X ( 2nd ` F ) z ) ` g ) ` A ) e. ( ( 1st ` F ) ` z ) )
61		eqid 2622	. . . . . . . 8 |- ( g e. ( z ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) z ) ` g ) ` A ) ) = ( g e. ( z ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) z ) ` g ) ` A ) )
62	60, 61	fmptd 6385	. . . . . . 7 |- ( ( ph /\ z e. B ) -> ( g e. ( z ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) z ) ` g ) ` A ) ) : ( z ( Hom ` C ) X ) --> ( ( 1st ` F ) ` z ) )
63	12	adantr 481	. . . . . . . . 9 |- ( ( ph /\ z e. B ) -> C e. Cat )
64	1, 2, 63, 35, 40, 36	yon11 16904	. . . . . . . 8 |- ( ( ph /\ z e. B ) -> ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` z ) = ( z ( Hom ` C ) X ) )
65	64	feq2d 6031	. . . . . . 7 |- ( ( ph /\ z e. B ) -> ( ( g e. ( z ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) z ) ` g ) ` A ) ) : ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` z ) --> ( ( 1st ` F ) ` z ) <-> ( g e. ( z ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) z ) ` g ) ` A ) ) : ( z ( Hom ` C ) X ) --> ( ( 1st ` F ) ` z ) ) )
66	62, 65	mpbird 247	. . . . . 6 |- ( ( ph /\ z e. B ) -> ( g e. ( z ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) z ) ` g ) ` A ) ) : ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` z ) --> ( ( 1st ` F ) ` z ) )
67	1, 2, 12, 17, 4, 5, 45, 14	yon1cl 16903	. . . . . . . . . . 11 |- ( ph -> ( ( 1st ` Y ) ` X ) e. ( O Func S ) )
68		1st2ndbr 7217	. . . . . . . . . . 11 |- ( ( Rel ( O Func S ) /\ ( ( 1st ` Y ) ` X ) e. ( O Func S ) ) -> ( 1st ` ( ( 1st ` Y ) ` X ) ) ( O Func S ) ( 2nd ` ( ( 1st ` Y ) ` X ) ) )
69	31, 67, 68	sylancr 695	. . . . . . . . . 10 |- ( ph -> ( 1st ` ( ( 1st ` Y ) ` X ) ) ( O Func S ) ( 2nd ` ( ( 1st ` Y ) ` X ) ) )
70	28, 48, 69	funcf1 16526	. . . . . . . . 9 |- ( ph -> ( 1st ` ( ( 1st ` Y ) ` X ) ) : B --> ( Base ` S ) )
71	50	feq3d 6032	. . . . . . . . 9 |- ( ph -> ( ( 1st ` ( ( 1st ` Y ) ` X ) ) : B --> U <-> ( 1st ` ( ( 1st ` Y ) ` X ) ) : B --> ( Base ` S ) ) )
72	70, 71	mpbird 247	. . . . . . . 8 |- ( ph -> ( 1st ` ( ( 1st ` Y ) ` X ) ) : B --> U )
73	72	ffvelrnda 6359	. . . . . . 7 |- ( ( ph /\ z e. B ) -> ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` z ) e. U )
74	5, 46, 30, 73, 55	elsetchom 16731	. . . . . 6 |- ( ( ph /\ z e. B ) -> ( ( g e. ( z ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) z ) ` g ) ` A ) ) e. ( ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` z ) ( Hom ` S ) ( ( 1st ` F ) ` z ) ) <-> ( g e. ( z ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) z ) ` g ) ` A ) ) : ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` z ) --> ( ( 1st ` F ) ` z ) ) )
75	66, 74	mpbird 247	. . . . 5 |- ( ( ph /\ z e. B ) -> ( g e. ( z ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) z ) ` g ) ` A ) ) e. ( ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` z ) ( Hom ` S ) ( ( 1st ` F ) ` z ) ) )
76	75	ralrimiva 2966	. . . 4 |- ( ph -> A. z e. B ( g e. ( z ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) z ) ` g ) ` A ) ) e. ( ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` z ) ( Hom ` S ) ( ( 1st ` F ) ` z ) ) )
77		fvex 6201	. . . . . 6 |- ( Base ` C ) e. _V
78	2, 77	eqeltri 2697	. . . . 5 |- B e. _V
79		mptelixpg 7945	. . . . 5 |- ( B e. _V -> ( ( z e. B |-> ( g e. ( z ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) z ) ` g ) ` A ) ) ) e. X_ z e. B ( ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` z ) ( Hom ` S ) ( ( 1st ` F ) ` z ) ) <-> A. z e. B ( g e. ( z ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) z ) ` g ) ` A ) ) e. ( ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` z ) ( Hom ` S ) ( ( 1st ` F ) ` z ) ) ) )
80	78, 79	ax-mp 5	. . . 4 |- ( ( z e. B |-> ( g e. ( z ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) z ) ` g ) ` A ) ) ) e. X_ z e. B ( ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` z ) ( Hom ` S ) ( ( 1st ` F ) ` z ) ) <-> A. z e. B ( g e. ( z ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) z ) ` g ) ` A ) ) e. ( ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` z ) ( Hom ` S ) ( ( 1st ` F ) ` z ) ) )
81	76, 80	sylibr 224	. . 3 |- ( ph -> ( z e. B |-> ( g e. ( z ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) z ) ` g ) ` A ) ) ) e. X_ z e. B ( ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` z ) ( Hom ` S ) ( ( 1st ` F ) ` z ) ) )
82	27, 81	eqeltrd 2701	. 2 |- ( ph -> ( ( F N X ) ` A ) e. X_ z e. B ( ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` z ) ( Hom ` S ) ( ( 1st ` F ) ` z ) ) )
83	12	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) -> C e. Cat )
84	17	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) -> X e. B )
85		simpr1 1067	. . . . . . . . . 10 |- ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) -> z e. B )
86	1, 2, 83, 84, 40, 85	yon11 16904	. . . . . . . . 9 |- ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) -> ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` z ) = ( z ( Hom ` C ) X ) )
87	86	eleq2d 2687	. . . . . . . 8 |- ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) -> ( k e. ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` z ) <-> k e. ( z ( Hom ` C ) X ) ) )
88	87	biimpa 501	. . . . . . 7 |- ( ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) /\ k e. ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` z ) ) -> k e. ( z ( Hom ` C ) X ) )
89		eqid 2622	. . . . . . . . . . . 12 |- (comp ` O ) = (comp ` O )
90		eqid 2622	. . . . . . . . . . . 12 |- (comp ` S ) = (comp ` S )
91	33	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) -> ( 1st ` F ) ( O Func S ) ( 2nd ` F ) )
92	91	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) /\ k e. ( z ( Hom ` C ) X ) ) -> ( 1st ` F ) ( O Func S ) ( 2nd ` F ) )
93	84	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) /\ k e. ( z ( Hom ` C ) X ) ) -> X e. B )
94	85	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) /\ k e. ( z ( Hom ` C ) X ) ) -> z e. B )
95		simpr2 1068	. . . . . . . . . . . . 13 |- ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) -> w e. B )
96	95	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) /\ k e. ( z ( Hom ` C ) X ) ) -> w e. B )
97		simpr 477	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) /\ k e. ( z ( Hom ` C ) X ) ) -> k e. ( z ( Hom ` C ) X ) )
98	97, 41	syl6eleqr 2712	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) /\ k e. ( z ( Hom ` C ) X ) ) -> k e. ( X ( Hom ` O ) z ) )
99		simplr3 1105	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) /\ k e. ( z ( Hom ` C ) X ) ) -> h e. ( z ( Hom ` O ) w ) )
100	28, 29, 89, 90, 92, 93, 94, 96, 98, 99	funcco 16531	. . . . . . . . . . 11 |- ( ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) /\ k e. ( z ( Hom ` C ) X ) ) -> ( ( X ( 2nd ` F ) w ) ` ( h ( <. X , z >. (comp ` O ) w ) k ) ) = ( ( ( z ( 2nd ` F ) w ) ` h ) ( <. ( ( 1st ` F ) ` X ) , ( ( 1st ` F ) ` z ) >. (comp ` S ) ( ( 1st ` F ) ` w ) ) ( ( X ( 2nd ` F ) z ) ` k ) ) )
101		eqid 2622	. . . . . . . . . . . . 13 |- (comp ` C ) = (comp ` C )
102	2, 101, 4, 93, 94, 96	oppcco 16377	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) /\ k e. ( z ( Hom ` C ) X ) ) -> ( h ( <. X , z >. (comp ` O ) w ) k ) = ( k ( <. w , z >. (comp ` C ) X ) h ) )
103	102	fveq2d 6195	. . . . . . . . . . 11 |- ( ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) /\ k e. ( z ( Hom ` C ) X ) ) -> ( ( X ( 2nd ` F ) w ) ` ( h ( <. X , z >. (comp ` O ) w ) k ) ) = ( ( X ( 2nd ` F ) w ) ` ( k ( <. w , z >. (comp ` C ) X ) h ) ) )
104	45	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) -> U e. _V )
105	104	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) /\ k e. ( z ( Hom ` C ) X ) ) -> U e. _V )
106	53	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) /\ k e. ( z ( Hom ` C ) X ) ) -> ( ( 1st ` F ) ` X ) e. U )
107	55	3ad2antr1 1226	. . . . . . . . . . . . 13 |- ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) -> ( ( 1st ` F ) ` z ) e. U )
108	107	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) /\ k e. ( z ( Hom ` C ) X ) ) -> ( ( 1st ` F ) ` z ) e. U )
109	52	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) -> ( 1st ` F ) : B --> U )
110	109, 95	ffvelrnd 6360	. . . . . . . . . . . . 13 |- ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) -> ( ( 1st ` F ) ` w ) e. U )
111	110	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) /\ k e. ( z ( Hom ` C ) X ) ) -> ( ( 1st ` F ) ` w ) e. U )
112	28, 29, 30, 91, 84, 85	funcf2 16528	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) -> ( X ( 2nd ` F ) z ) : ( X ( Hom ` O ) z ) --> ( ( ( 1st ` F ) ` X ) ( Hom ` S ) ( ( 1st ` F ) ` z ) ) )
113	112	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) /\ k e. ( z ( Hom ` C ) X ) ) -> ( X ( 2nd ` F ) z ) : ( X ( Hom ` O ) z ) --> ( ( ( 1st ` F ) ` X ) ( Hom ` S ) ( ( 1st ` F ) ` z ) ) )
114	113, 98	ffvelrnd 6360	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) /\ k e. ( z ( Hom ` C ) X ) ) -> ( ( X ( 2nd ` F ) z ) ` k ) e. ( ( ( 1st ` F ) ` X ) ( Hom ` S ) ( ( 1st ` F ) ` z ) ) )
115	5, 105, 30, 106, 108	elsetchom 16731	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) /\ k e. ( z ( Hom ` C ) X ) ) -> ( ( ( X ( 2nd ` F ) z ) ` k ) e. ( ( ( 1st ` F ) ` X ) ( Hom ` S ) ( ( 1st ` F ) ` z ) ) <-> ( ( X ( 2nd ` F ) z ) ` k ) : ( ( 1st ` F ) ` X ) --> ( ( 1st ` F ) ` z ) ) )
116	114, 115	mpbid 222	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) /\ k e. ( z ( Hom ` C ) X ) ) -> ( ( X ( 2nd ` F ) z ) ` k ) : ( ( 1st ` F ) ` X ) --> ( ( 1st ` F ) ` z ) )
117	28, 29, 30, 91, 85, 95	funcf2 16528	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) -> ( z ( 2nd ` F ) w ) : ( z ( Hom ` O ) w ) --> ( ( ( 1st ` F ) ` z ) ( Hom ` S ) ( ( 1st ` F ) ` w ) ) )
118		simpr3 1069	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) -> h e. ( z ( Hom ` O ) w ) )
119	117, 118	ffvelrnd 6360	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) -> ( ( z ( 2nd ` F ) w ) ` h ) e. ( ( ( 1st ` F ) ` z ) ( Hom ` S ) ( ( 1st ` F ) ` w ) ) )
120	5, 104, 30, 107, 110	elsetchom 16731	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) -> ( ( ( z ( 2nd ` F ) w ) ` h ) e. ( ( ( 1st ` F ) ` z ) ( Hom ` S ) ( ( 1st ` F ) ` w ) ) <-> ( ( z ( 2nd ` F ) w ) ` h ) : ( ( 1st ` F ) ` z ) --> ( ( 1st ` F ) ` w ) ) )
121	119, 120	mpbid 222	. . . . . . . . . . . . 13 |- ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) -> ( ( z ( 2nd ` F ) w ) ` h ) : ( ( 1st ` F ) ` z ) --> ( ( 1st ` F ) ` w ) )
122	121	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) /\ k e. ( z ( Hom ` C ) X ) ) -> ( ( z ( 2nd ` F ) w ) ` h ) : ( ( 1st ` F ) ` z ) --> ( ( 1st ` F ) ` w ) )
123	5, 105, 90, 106, 108, 111, 116, 122	setcco 16733	. . . . . . . . . . 11 |- ( ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) /\ k e. ( z ( Hom ` C ) X ) ) -> ( ( ( z ( 2nd ` F ) w ) ` h ) ( <. ( ( 1st ` F ) ` X ) , ( ( 1st ` F ) ` z ) >. (comp ` S ) ( ( 1st ` F ) ` w ) ) ( ( X ( 2nd ` F ) z ) ` k ) ) = ( ( ( z ( 2nd ` F ) w ) ` h ) o. ( ( X ( 2nd ` F ) z ) ` k ) ) )
124	100, 103, 123	3eqtr3d 2664	. . . . . . . . . 10 |- ( ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) /\ k e. ( z ( Hom ` C ) X ) ) -> ( ( X ( 2nd ` F ) w ) ` ( k ( <. w , z >. (comp ` C ) X ) h ) ) = ( ( ( z ( 2nd ` F ) w ) ` h ) o. ( ( X ( 2nd ` F ) z ) ` k ) ) )
125	124	fveq1d 6193	. . . . . . . . 9 |- ( ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) /\ k e. ( z ( Hom ` C ) X ) ) -> ( ( ( X ( 2nd ` F ) w ) ` ( k ( <. w , z >. (comp ` C ) X ) h ) ) ` A ) = ( ( ( ( z ( 2nd ` F ) w ) ` h ) o. ( ( X ( 2nd ` F ) z ) ` k ) ) ` A ) )
126	19	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) /\ k e. ( z ( Hom ` C ) X ) ) -> A e. ( ( 1st ` F ) ` X ) )
127		fvco3 6275	. . . . . . . . . 10 |- ( ( ( ( X ( 2nd ` F ) z ) ` k ) : ( ( 1st ` F ) ` X ) --> ( ( 1st ` F ) ` z ) /\ A e. ( ( 1st ` F ) ` X ) ) -> ( ( ( ( z ( 2nd ` F ) w ) ` h ) o. ( ( X ( 2nd ` F ) z ) ` k ) ) ` A ) = ( ( ( z ( 2nd ` F ) w ) ` h ) ` ( ( ( X ( 2nd ` F ) z ) ` k ) ` A ) ) )
128	116, 126, 127	syl2anc 693	. . . . . . . . 9 |- ( ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) /\ k e. ( z ( Hom ` C ) X ) ) -> ( ( ( ( z ( 2nd ` F ) w ) ` h ) o. ( ( X ( 2nd ` F ) z ) ` k ) ) ` A ) = ( ( ( z ( 2nd ` F ) w ) ` h ) ` ( ( ( X ( 2nd ` F ) z ) ` k ) ` A ) ) )
129	125, 128	eqtrd 2656	. . . . . . . 8 |- ( ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) /\ k e. ( z ( Hom ` C ) X ) ) -> ( ( ( X ( 2nd ` F ) w ) ` ( k ( <. w , z >. (comp ` C ) X ) h ) ) ` A ) = ( ( ( z ( 2nd ` F ) w ) ` h ) ` ( ( ( X ( 2nd ` F ) z ) ` k ) ` A ) ) )
130	83	adantr 481	. . . . . . . . . . 11 |- ( ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) /\ k e. ( z ( Hom ` C ) X ) ) -> C e. Cat )
131	40, 4	oppchom 16375	. . . . . . . . . . . 12 |- ( z ( Hom ` O ) w ) = ( w ( Hom ` C ) z )
132	99, 131	syl6eleq 2711	. . . . . . . . . . 11 |- ( ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) /\ k e. ( z ( Hom ` C ) X ) ) -> h e. ( w ( Hom ` C ) z ) )
133	1, 2, 130, 93, 40, 94, 101, 96, 132, 97	yon12 16905	. . . . . . . . . 10 |- ( ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) /\ k e. ( z ( Hom ` C ) X ) ) -> ( ( ( z ( 2nd ` ( ( 1st ` Y ) ` X ) ) w ) ` h ) ` k ) = ( k ( <. w , z >. (comp ` C ) X ) h ) )
134	133	fveq2d 6195	. . . . . . . . 9 |- ( ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) /\ k e. ( z ( Hom ` C ) X ) ) -> ( ( ( ( F N X ) ` A ) ` w ) ` ( ( ( z ( 2nd ` ( ( 1st ` Y ) ` X ) ) w ) ` h ) ` k ) ) = ( ( ( ( F N X ) ` A ) ` w ) ` ( k ( <. w , z >. (comp ` C ) X ) h ) ) )
135	13	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) /\ k e. ( z ( Hom ` C ) X ) ) -> V e. W )
136	14	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) /\ k e. ( z ( Hom ` C ) X ) ) -> ran ( Hom f ` C ) C_ U )
137	15	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) /\ k e. ( z ( Hom ` C ) X ) ) -> ( ran ( Hom f ` Q ) u. U ) C_ V )
138	16	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) /\ k e. ( z ( Hom ` C ) X ) ) -> F e. ( O Func S ) )
139	2, 40, 101, 130, 96, 94, 93, 132, 97	catcocl 16346	. . . . . . . . . 10 |- ( ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) /\ k e. ( z ( Hom ` C ) X ) ) -> ( k ( <. w , z >. (comp ` C ) X ) h ) e. ( w ( Hom ` C ) X ) )
140	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 130, 135, 136, 137, 138, 93, 18, 126, 96, 139	yonedalem4b 16916	. . . . . . . . 9 |- ( ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) /\ k e. ( z ( Hom ` C ) X ) ) -> ( ( ( ( F N X ) ` A ) ` w ) ` ( k ( <. w , z >. (comp ` C ) X ) h ) ) = ( ( ( X ( 2nd ` F ) w ) ` ( k ( <. w , z >. (comp ` C ) X ) h ) ) ` A ) )
141	134, 140	eqtrd 2656	. . . . . . . 8 |- ( ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) /\ k e. ( z ( Hom ` C ) X ) ) -> ( ( ( ( F N X ) ` A ) ` w ) ` ( ( ( z ( 2nd ` ( ( 1st ` Y ) ` X ) ) w ) ` h ) ` k ) ) = ( ( ( X ( 2nd ` F ) w ) ` ( k ( <. w , z >. (comp ` C ) X ) h ) ) ` A ) )
142	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 130, 135, 136, 137, 138, 93, 18, 126, 94, 97	yonedalem4b 16916	. . . . . . . . 9 |- ( ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) /\ k e. ( z ( Hom ` C ) X ) ) -> ( ( ( ( F N X ) ` A ) ` z ) ` k ) = ( ( ( X ( 2nd ` F ) z ) ` k ) ` A ) )
143	142	fveq2d 6195	. . . . . . . 8 |- ( ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) /\ k e. ( z ( Hom ` C ) X ) ) -> ( ( ( z ( 2nd ` F ) w ) ` h ) ` ( ( ( ( F N X ) ` A ) ` z ) ` k ) ) = ( ( ( z ( 2nd ` F ) w ) ` h ) ` ( ( ( X ( 2nd ` F ) z ) ` k ) ` A ) ) )
144	129, 141, 143	3eqtr4d 2666	. . . . . . 7 |- ( ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) /\ k e. ( z ( Hom ` C ) X ) ) -> ( ( ( ( F N X ) ` A ) ` w ) ` ( ( ( z ( 2nd ` ( ( 1st ` Y ) ` X ) ) w ) ` h ) ` k ) ) = ( ( ( z ( 2nd ` F ) w ) ` h ) ` ( ( ( ( F N X ) ` A ) ` z ) ` k ) ) )
145	88, 144	syldan 487	. . . . . 6 |- ( ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) /\ k e. ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` z ) ) -> ( ( ( ( F N X ) ` A ) ` w ) ` ( ( ( z ( 2nd ` ( ( 1st ` Y ) ` X ) ) w ) ` h ) ` k ) ) = ( ( ( z ( 2nd ` F ) w ) ` h ) ` ( ( ( ( F N X ) ` A ) ` z ) ` k ) ) )
146	145	mpteq2dva 4744	. . . . 5 |- ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) -> ( k e. ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` z ) |-> ( ( ( ( F N X ) ` A ) ` w ) ` ( ( ( z ( 2nd ` ( ( 1st ` Y ) ` X ) ) w ) ` h ) ` k ) ) ) = ( k e. ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` z ) |-> ( ( ( z ( 2nd ` F ) w ) ` h ) ` ( ( ( ( F N X ) ` A ) ` z ) ` k ) ) ) )
147	27	fveq1d 6193	. . . . . . . . . . . 12 |- ( ph -> ( ( ( F N X ) ` A ) ` z ) = ( ( z e. B |-> ( g e. ( z ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) z ) ` g ) ` A ) ) ) ` z ) )
148		ovex 6678	. . . . . . . . . . . . . 14 |- ( z ( Hom ` C ) X ) e. _V
149	148	mptex 6486	. . . . . . . . . . . . 13 |- ( g e. ( z ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) z ) ` g ) ` A ) ) e. _V
150		eqid 2622	. . . . . . . . . . . . . 14 |- ( z e. B |-> ( g e. ( z ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) z ) ` g ) ` A ) ) ) = ( z e. B |-> ( g e. ( z ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) z ) ` g ) ` A ) ) )
151	150	fvmpt2 6291	. . . . . . . . . . . . 13 |- ( ( z e. B /\ ( g e. ( z ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) z ) ` g ) ` A ) ) e. _V ) -> ( ( z e. B |-> ( g e. ( z ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) z ) ` g ) ` A ) ) ) ` z ) = ( g e. ( z ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) z ) ` g ) ` A ) ) )
152	149, 151	mpan2 707	. . . . . . . . . . . 12 |- ( z e. B -> ( ( z e. B |-> ( g e. ( z ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) z ) ` g ) ` A ) ) ) ` z ) = ( g e. ( z ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) z ) ` g ) ` A ) ) )
153	147, 152	sylan9eq 2676	. . . . . . . . . . 11 |- ( ( ph /\ z e. B ) -> ( ( ( F N X ) ` A ) ` z ) = ( g e. ( z ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) z ) ` g ) ` A ) ) )
154	153	feq1d 6030	. . . . . . . . . 10 |- ( ( ph /\ z e. B ) -> ( ( ( ( F N X ) ` A ) ` z ) : ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` z ) --> ( ( 1st ` F ) ` z ) <-> ( g e. ( z ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) z ) ` g ) ` A ) ) : ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` z ) --> ( ( 1st ` F ) ` z ) ) )
155	66, 154	mpbird 247	. . . . . . . . 9 |- ( ( ph /\ z e. B ) -> ( ( ( F N X ) ` A ) ` z ) : ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` z ) --> ( ( 1st ` F ) ` z ) )
156	155	ralrimiva 2966	. . . . . . . 8 |- ( ph -> A. z e. B ( ( ( F N X ) ` A ) ` z ) : ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` z ) --> ( ( 1st ` F ) ` z ) )
157	156	adantr 481	. . . . . . 7 |- ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) -> A. z e. B ( ( ( F N X ) ` A ) ` z ) : ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` z ) --> ( ( 1st ` F ) ` z ) )
158		fveq2 6191	. . . . . . . . 9 |- ( z = w -> ( ( ( F N X ) ` A ) ` z ) = ( ( ( F N X ) ` A ) ` w ) )
159		fveq2 6191	. . . . . . . . 9 |- ( z = w -> ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` z ) = ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` w ) )
160		fveq2 6191	. . . . . . . . 9 |- ( z = w -> ( ( 1st ` F ) ` z ) = ( ( 1st ` F ) ` w ) )
161	158, 159, 160	feq123d 6034	. . . . . . . 8 |- ( z = w -> ( ( ( ( F N X ) ` A ) ` z ) : ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` z ) --> ( ( 1st ` F ) ` z ) <-> ( ( ( F N X ) ` A ) ` w ) : ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` w ) --> ( ( 1st ` F ) ` w ) ) )
162	161	rspcv 3305	. . . . . . 7 |- ( w e. B -> ( A. z e. B ( ( ( F N X ) ` A ) ` z ) : ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` z ) --> ( ( 1st ` F ) ` z ) -> ( ( ( F N X ) ` A ) ` w ) : ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` w ) --> ( ( 1st ` F ) ` w ) ) )
163	95, 157, 162	sylc 65	. . . . . 6 |- ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) -> ( ( ( F N X ) ` A ) ` w ) : ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` w ) --> ( ( 1st ` F ) ` w ) )
164	69	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) -> ( 1st ` ( ( 1st ` Y ) ` X ) ) ( O Func S ) ( 2nd ` ( ( 1st ` Y ) ` X ) ) )
165	28, 29, 30, 164, 85, 95	funcf2 16528	. . . . . . . 8 |- ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) -> ( z ( 2nd ` ( ( 1st ` Y ) ` X ) ) w ) : ( z ( Hom ` O ) w ) --> ( ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` z ) ( Hom ` S ) ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` w ) ) )
166	165, 118	ffvelrnd 6360	. . . . . . 7 |- ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) -> ( ( z ( 2nd ` ( ( 1st ` Y ) ` X ) ) w ) ` h ) e. ( ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` z ) ( Hom ` S ) ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` w ) ) )
167	73	3ad2antr1 1226	. . . . . . . 8 |- ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) -> ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` z ) e. U )
168	72	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) -> ( 1st ` ( ( 1st ` Y ) ` X ) ) : B --> U )
169	168, 95	ffvelrnd 6360	. . . . . . . 8 |- ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) -> ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` w ) e. U )
170	5, 104, 30, 167, 169	elsetchom 16731	. . . . . . 7 |- ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) -> ( ( ( z ( 2nd ` ( ( 1st ` Y ) ` X ) ) w ) ` h ) e. ( ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` z ) ( Hom ` S ) ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` w ) ) <-> ( ( z ( 2nd ` ( ( 1st ` Y ) ` X ) ) w ) ` h ) : ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` z ) --> ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` w ) ) )
171	166, 170	mpbid 222	. . . . . 6 |- ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) -> ( ( z ( 2nd ` ( ( 1st ` Y ) ` X ) ) w ) ` h ) : ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` z ) --> ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` w ) )
172		fcompt 6400	. . . . . 6 |- ( ( ( ( ( F N X ) ` A ) ` w ) : ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` w ) --> ( ( 1st ` F ) ` w ) /\ ( ( z ( 2nd ` ( ( 1st ` Y ) ` X ) ) w ) ` h ) : ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` z ) --> ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` w ) ) -> ( ( ( ( F N X ) ` A ) ` w ) o. ( ( z ( 2nd ` ( ( 1st ` Y ) ` X ) ) w ) ` h ) ) = ( k e. ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` z ) |-> ( ( ( ( F N X ) ` A ) ` w ) ` ( ( ( z ( 2nd ` ( ( 1st ` Y ) ` X ) ) w ) ` h ) ` k ) ) ) )
173	163, 171, 172	syl2anc 693	. . . . 5 |- ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) -> ( ( ( ( F N X ) ` A ) ` w ) o. ( ( z ( 2nd ` ( ( 1st ` Y ) ` X ) ) w ) ` h ) ) = ( k e. ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` z ) |-> ( ( ( ( F N X ) ` A ) ` w ) ` ( ( ( z ( 2nd ` ( ( 1st ` Y ) ` X ) ) w ) ` h ) ` k ) ) ) )
174	155	3ad2antr1 1226	. . . . . 6 |- ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) -> ( ( ( F N X ) ` A ) ` z ) : ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` z ) --> ( ( 1st ` F ) ` z ) )
175		fcompt 6400	. . . . . 6 |- ( ( ( ( z ( 2nd ` F ) w ) ` h ) : ( ( 1st ` F ) ` z ) --> ( ( 1st ` F ) ` w ) /\ ( ( ( F N X ) ` A ) ` z ) : ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` z ) --> ( ( 1st ` F ) ` z ) ) -> ( ( ( z ( 2nd ` F ) w ) ` h ) o. ( ( ( F N X ) ` A ) ` z ) ) = ( k e. ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` z ) |-> ( ( ( z ( 2nd ` F ) w ) ` h ) ` ( ( ( ( F N X ) ` A ) ` z ) ` k ) ) ) )
176	121, 174, 175	syl2anc 693	. . . . 5 |- ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) -> ( ( ( z ( 2nd ` F ) w ) ` h ) o. ( ( ( F N X ) ` A ) ` z ) ) = ( k e. ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` z ) |-> ( ( ( z ( 2nd ` F ) w ) ` h ) ` ( ( ( ( F N X ) ` A ) ` z ) ` k ) ) ) )
177	146, 173, 176	3eqtr4d 2666	. . . 4 |- ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) -> ( ( ( ( F N X ) ` A ) ` w ) o. ( ( z ( 2nd ` ( ( 1st ` Y ) ` X ) ) w ) ` h ) ) = ( ( ( z ( 2nd ` F ) w ) ` h ) o. ( ( ( F N X ) ` A ) ` z ) ) )
178	5, 104, 90, 167, 169, 110, 171, 163	setcco 16733	. . . 4 |- ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) -> ( ( ( ( F N X ) ` A ) ` w ) ( <. ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` z ) , ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` w ) >. (comp ` S ) ( ( 1st ` F ) ` w ) ) ( ( z ( 2nd ` ( ( 1st ` Y ) ` X ) ) w ) ` h ) ) = ( ( ( ( F N X ) ` A ) ` w ) o. ( ( z ( 2nd ` ( ( 1st ` Y ) ` X ) ) w ) ` h ) ) )
179	5, 104, 90, 167, 107, 110, 174, 121	setcco 16733	. . . 4 |- ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) -> ( ( ( z ( 2nd ` F ) w ) ` h ) ( <. ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` z ) , ( ( 1st ` F ) ` z ) >. (comp ` S ) ( ( 1st ` F ) ` w ) ) ( ( ( F N X ) ` A ) ` z ) ) = ( ( ( z ( 2nd ` F ) w ) ` h ) o. ( ( ( F N X ) ` A ) ` z ) ) )
180	177, 178, 179	3eqtr4d 2666	. . 3 |- ( ( ph /\ ( z e. B /\ w e. B /\ h e. ( z ( Hom ` O ) w ) ) ) -> ( ( ( ( F N X ) ` A ) ` w ) ( <. ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` z ) , ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` w ) >. (comp ` S ) ( ( 1st ` F ) ` w ) ) ( ( z ( 2nd ` ( ( 1st ` Y ) ` X ) ) w ) ` h ) ) = ( ( ( z ( 2nd ` F ) w ) ` h ) ( <. ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` z ) , ( ( 1st ` F ) ` z ) >. (comp ` S ) ( ( 1st ` F ) ` w ) ) ( ( ( F N X ) ` A ) ` z ) ) )
181	180	ralrimivvva 2972	. 2 |- ( ph -> A. z e. B A. w e. B A. h e. ( z ( Hom ` O ) w ) ( ( ( ( F N X ) ` A ) ` w ) ( <. ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` z ) , ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` w ) >. (comp ` S ) ( ( 1st ` F ) ` w ) ) ( ( z ( 2nd ` ( ( 1st ` Y ) ` X ) ) w ) ` h ) ) = ( ( ( z ( 2nd ` F ) w ) ` h ) ( <. ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` z ) , ( ( 1st ` F ) ` z ) >. (comp ` S ) ( ( 1st ` F ) ` w ) ) ( ( ( F N X ) ` A ) ` z ) ) )
182		eqid 2622	. . 3 |- ( O Nat S ) = ( O Nat S )
183	182, 28, 29, 30, 90, 67, 16	isnat2 16608	. 2 |- ( ph -> ( ( ( F N X ) ` A ) e. ( ( ( 1st ` Y ) ` X ) ( O Nat S ) F ) <-> ( ( ( F N X ) ` A ) e. X_ z e. B ( ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` z ) ( Hom ` S ) ( ( 1st ` F ) ` z ) ) /\ A. z e. B A. w e. B A. h e. ( z ( Hom ` O ) w ) ( ( ( ( F N X ) ` A ) ` w ) ( <. ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` z ) , ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` w ) >. (comp ` S ) ( ( 1st ` F ) ` w ) ) ( ( z ( 2nd ` ( ( 1st ` Y ) ` X ) ) w ) ` h ) ) = ( ( ( z ( 2nd ` F ) w ) ` h ) ( <. ( ( 1st ` ( ( 1st ` Y ) ` X ) ) ` z ) , ( ( 1st ` F ) ` z ) >. (comp ` S ) ( ( 1st ` F ) ` w ) ) ( ( ( F N X ) ` A ) ` z ) ) ) ) )
184	82, 181, 183	mpbir2and 957	1 |- ( ph -> ( ( F N X ) ` A ) e. ( ( ( 1st ` Y ) ` X ) ( O Nat S ) F ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   u. cun 3572   C_ wss 3574   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   ran crn 5115   o. ccom 5118   Rel wrel 5119   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167  tpos ctpos 7351   X_cixp 7908   Basecbs 15857   Hom chom 15952  compcco 15953   Catccat 16325   Idccid 16326   Hom _f chomf 16327  oppCatcoppc 16371   Func cfunc 16514   o._func ccofu 16516   Nat cnat 16601   FuncCat cfuc 16602   SetCatcsetc 16725   X._c cxpc 16808   1st_F c1stf 16809   2nd_F c2ndf 16810   ⟨,⟩_F cprf 16811   eval_F cevlf 16849  Hom_Fchof 16888  Yoncyon 16889

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-fal 1489  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-tpos 7352  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-sets 15864  df-hom 15966  df-cco 15967  df-cat 16329  df-cid 16330  df-homf 16331  df-comf 16332  df-oppc 16372  df-func 16518  df-nat 16603  df-fuc 16604  df-setc 16726  df-xpc 16812  df-curf 16854  df-hof 16890  df-yon 16891

This theorem is referenced by:  yonedainv  16921