Theorem yonedalem3 16920

Description: Lemma for yoneda 16923. (Contributed by Mario Carneiro, 28-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 )
yoneda.m	|- M = ( f e. ( O Func S ) , x e. B |-> ( a e. ( ( ( 1st ` Y ) ` x ) ( O Nat S ) f ) |-> ( ( a ` x ) ` ( .1. ` x ) ) ) )

Assertion

Ref	Expression
yonedalem3	|- ( ph -> M e. ( Z ( ( Q X.c O ) Nat T ) E ) )

Distinct variable groups:   f, a, x, .1.   C, a, f, x   E, a, f   B, a, f, x   O, a, f, x   S, a, f, x   Q, a, f, x   T, f   ph, a, f, x   Y, a, f, x   Z, a, f, x

Allowed substitution hints:   R( x, f, a)   T( x, a)   U( x, f, a)   E( x)   H( x, f, a)   M( x, f, a)   V( x, f, a)   W( x, f, a)

Proof of Theorem yonedalem3

Dummy variables g y w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		yoneda.m	. . . . 5 |- M = ( f e. ( O Func S ) , x e. B |-> ( a e. ( ( ( 1st ` Y ) ` x ) ( O Nat S ) f ) |-> ( ( a ` x ) ` ( .1. ` x ) ) ) )
2		ovex 6678	. . . . . 6 |- ( ( ( 1st ` Y ) ` x ) ( O Nat S ) f ) e. _V
3	2	mptex 6486	. . . . 5 |- ( a e. ( ( ( 1st ` Y ) ` x ) ( O Nat S ) f ) |-> ( ( a ` x ) ` ( .1. ` x ) ) ) e. _V
4	1, 3	fnmpt2i 7239	. . . 4 |- M Fn ( ( O Func S ) X. B )
5	4	a1i 11	. . 3 |- ( ph -> M Fn ( ( O Func S ) X. B ) )
6		yoneda.y	. . . . . . . 8 |- Y = (Yon ` C )
7		yoneda.b	. . . . . . . 8 |- B = ( Base ` C )
8		yoneda.1	. . . . . . . 8 |- .1. = ( Id ` C )
9		yoneda.o	. . . . . . . 8 |- O = (oppCat ` C )
10		yoneda.s	. . . . . . . 8 |- S = ( SetCat ` U )
11		yoneda.t	. . . . . . . 8 |- T = ( SetCat ` V )
12		yoneda.q	. . . . . . . 8 |- Q = ( O FuncCat S )
13		yoneda.h	. . . . . . . 8 |- H = (HomF ` Q )
14		yoneda.r	. . . . . . . 8 |- R = ( ( Q X.c O ) FuncCat T )
15		yoneda.e	. . . . . . . 8 |- E = ( O evalF S )
16		yoneda.z	. . . . . . . 8 |- Z = ( H o.func ( ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ⟨,⟩F ( Q 1stF O ) ) )
17		yoneda.c	. . . . . . . . 9 |- ( ph -> C e. Cat )
18	17	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( g e. ( O Func S ) /\ y e. B ) ) -> C e. Cat )
19		yoneda.w	. . . . . . . . 9 |- ( ph -> V e. W )
20	19	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( g e. ( O Func S ) /\ y e. B ) ) -> V e. W )
21		yoneda.u	. . . . . . . . 9 |- ( ph -> ran ( Hom f ` C ) C_ U )
22	21	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( g e. ( O Func S ) /\ y e. B ) ) -> ran ( Hom f ` C ) C_ U )
23		yoneda.v	. . . . . . . . 9 |- ( ph -> ( ran ( Hom f ` Q ) u. U ) C_ V )
24	23	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( g e. ( O Func S ) /\ y e. B ) ) -> ( ran ( Hom f ` Q ) u. U ) C_ V )
25		simprl 794	. . . . . . . 8 |- ( ( ph /\ ( g e. ( O Func S ) /\ y e. B ) ) -> g e. ( O Func S ) )
26		simprr 796	. . . . . . . 8 |- ( ( ph /\ ( g e. ( O Func S ) /\ y e. B ) ) -> y e. B )
27	6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 25, 26, 1	yonedalem3a 16914	. . . . . . 7 |- ( ( ph /\ ( g e. ( O Func S ) /\ y e. B ) ) -> ( ( g M y ) = ( a e. ( ( ( 1st ` Y ) ` y ) ( O Nat S ) g ) |-> ( ( a ` y ) ` ( .1. ` y ) ) ) /\ ( g M y ) : ( g ( 1st ` Z ) y ) --> ( g ( 1st ` E ) y ) ) )
28	27	simprd 479	. . . . . 6 |- ( ( ph /\ ( g e. ( O Func S ) /\ y e. B ) ) -> ( g M y ) : ( g ( 1st ` Z ) y ) --> ( g ( 1st ` E ) y ) )
29		eqid 2622	. . . . . . 7 |- ( Hom ` T ) = ( Hom ` T )
30		eqid 2622	. . . . . . . . . . 11 |- ( Q X.c O ) = ( Q X.c O )
31	12	fucbas 16620	. . . . . . . . . . 11 |- ( O Func S ) = ( Base ` Q )
32	9, 7	oppcbas 16378	. . . . . . . . . . 11 |- B = ( Base ` O )
33	30, 31, 32	xpcbas 16818	. . . . . . . . . 10 |- ( ( O Func S ) X. B ) = ( Base ` ( Q X.c O ) )
34		eqid 2622	. . . . . . . . . 10 |- ( Base ` T ) = ( Base ` T )
35		relfunc 16522	. . . . . . . . . . 11 |- Rel ( ( Q X.c O ) Func T )
36	6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 21, 23	yonedalem1 16912	. . . . . . . . . . . 12 |- ( ph -> ( Z e. ( ( Q X.c O ) Func T ) /\ E e. ( ( Q X.c O ) Func T ) ) )
37	36	simpld 475	. . . . . . . . . . 11 |- ( ph -> Z e. ( ( Q X.c O ) Func T ) )
38		1st2ndbr 7217	. . . . . . . . . . 11 |- ( ( Rel ( ( Q X.c O ) Func T ) /\ Z e. ( ( Q X.c O ) Func T ) ) -> ( 1st ` Z ) ( ( Q X.c O ) Func T ) ( 2nd ` Z ) )
39	35, 37, 38	sylancr 695	. . . . . . . . . 10 |- ( ph -> ( 1st ` Z ) ( ( Q X.c O ) Func T ) ( 2nd ` Z ) )
40	33, 34, 39	funcf1 16526	. . . . . . . . 9 |- ( ph -> ( 1st ` Z ) : ( ( O Func S ) X. B ) --> ( Base ` T ) )
41	40	fovrnda 6805	. . . . . . . 8 |- ( ( ph /\ ( g e. ( O Func S ) /\ y e. B ) ) -> ( g ( 1st ` Z ) y ) e. ( Base ` T ) )
42	11, 20	setcbas 16728	. . . . . . . 8 |- ( ( ph /\ ( g e. ( O Func S ) /\ y e. B ) ) -> V = ( Base ` T ) )
43	41, 42	eleqtrrd 2704	. . . . . . 7 |- ( ( ph /\ ( g e. ( O Func S ) /\ y e. B ) ) -> ( g ( 1st ` Z ) y ) e. V )
44	36	simprd 479	. . . . . . . . . . 11 |- ( ph -> E e. ( ( Q X.c O ) Func T ) )
45		1st2ndbr 7217	. . . . . . . . . . 11 |- ( ( Rel ( ( Q X.c O ) Func T ) /\ E e. ( ( Q X.c O ) Func T ) ) -> ( 1st ` E ) ( ( Q X.c O ) Func T ) ( 2nd ` E ) )
46	35, 44, 45	sylancr 695	. . . . . . . . . 10 |- ( ph -> ( 1st ` E ) ( ( Q X.c O ) Func T ) ( 2nd ` E ) )
47	33, 34, 46	funcf1 16526	. . . . . . . . 9 |- ( ph -> ( 1st ` E ) : ( ( O Func S ) X. B ) --> ( Base ` T ) )
48	47	fovrnda 6805	. . . . . . . 8 |- ( ( ph /\ ( g e. ( O Func S ) /\ y e. B ) ) -> ( g ( 1st ` E ) y ) e. ( Base ` T ) )
49	48, 42	eleqtrrd 2704	. . . . . . 7 |- ( ( ph /\ ( g e. ( O Func S ) /\ y e. B ) ) -> ( g ( 1st ` E ) y ) e. V )
50	11, 20, 29, 43, 49	elsetchom 16731	. . . . . 6 |- ( ( ph /\ ( g e. ( O Func S ) /\ y e. B ) ) -> ( ( g M y ) e. ( ( g ( 1st ` Z ) y ) ( Hom ` T ) ( g ( 1st ` E ) y ) ) <-> ( g M y ) : ( g ( 1st ` Z ) y ) --> ( g ( 1st ` E ) y ) ) )
51	28, 50	mpbird 247	. . . . 5 |- ( ( ph /\ ( g e. ( O Func S ) /\ y e. B ) ) -> ( g M y ) e. ( ( g ( 1st ` Z ) y ) ( Hom ` T ) ( g ( 1st ` E ) y ) ) )
52	51	ralrimivva 2971	. . . 4 |- ( ph -> A. g e. ( O Func S ) A. y e. B ( g M y ) e. ( ( g ( 1st ` Z ) y ) ( Hom ` T ) ( g ( 1st ` E ) y ) ) )
53		fveq2 6191	. . . . . . 7 |- ( z = <. g , y >. -> ( M ` z ) = ( M ` <. g , y >. ) )
54		df-ov 6653	. . . . . . 7 |- ( g M y ) = ( M ` <. g , y >. )
55	53, 54	syl6eqr 2674	. . . . . 6 |- ( z = <. g , y >. -> ( M ` z ) = ( g M y ) )
56		fveq2 6191	. . . . . . . 8 |- ( z = <. g , y >. -> ( ( 1st ` Z ) ` z ) = ( ( 1st ` Z ) ` <. g , y >. ) )
57		df-ov 6653	. . . . . . . 8 |- ( g ( 1st ` Z ) y ) = ( ( 1st ` Z ) ` <. g , y >. )
58	56, 57	syl6eqr 2674	. . . . . . 7 |- ( z = <. g , y >. -> ( ( 1st ` Z ) ` z ) = ( g ( 1st ` Z ) y ) )
59		fveq2 6191	. . . . . . . 8 |- ( z = <. g , y >. -> ( ( 1st ` E ) ` z ) = ( ( 1st ` E ) ` <. g , y >. ) )
60		df-ov 6653	. . . . . . . 8 |- ( g ( 1st ` E ) y ) = ( ( 1st ` E ) ` <. g , y >. )
61	59, 60	syl6eqr 2674	. . . . . . 7 |- ( z = <. g , y >. -> ( ( 1st ` E ) ` z ) = ( g ( 1st ` E ) y ) )
62	58, 61	oveq12d 6668	. . . . . 6 |- ( z = <. g , y >. -> ( ( ( 1st ` Z ) ` z ) ( Hom ` T ) ( ( 1st ` E ) ` z ) ) = ( ( g ( 1st ` Z ) y ) ( Hom ` T ) ( g ( 1st ` E ) y ) ) )
63	55, 62	eleq12d 2695	. . . . 5 |- ( z = <. g , y >. -> ( ( M ` z ) e. ( ( ( 1st ` Z ) ` z ) ( Hom ` T ) ( ( 1st ` E ) ` z ) ) <-> ( g M y ) e. ( ( g ( 1st ` Z ) y ) ( Hom ` T ) ( g ( 1st ` E ) y ) ) ) )
64	63	ralxp 5263	. . . 4 |- ( A. z e. ( ( O Func S ) X. B ) ( M ` z ) e. ( ( ( 1st ` Z ) ` z ) ( Hom ` T ) ( ( 1st ` E ) ` z ) ) <-> A. g e. ( O Func S ) A. y e. B ( g M y ) e. ( ( g ( 1st ` Z ) y ) ( Hom ` T ) ( g ( 1st ` E ) y ) ) )
65	52, 64	sylibr 224	. . 3 |- ( ph -> A. z e. ( ( O Func S ) X. B ) ( M ` z ) e. ( ( ( 1st ` Z ) ` z ) ( Hom ` T ) ( ( 1st ` E ) ` z ) ) )
66		ovex 6678	. . . . . 6 |- ( O Func S ) e. _V
67		fvex 6201	. . . . . . 7 |- ( Base ` C ) e. _V
68	7, 67	eqeltri 2697	. . . . . 6 |- B e. _V
69	66, 68	mpt2ex 7247	. . . . 5 |- ( f e. ( O Func S ) , x e. B |-> ( a e. ( ( ( 1st ` Y ) ` x ) ( O Nat S ) f ) |-> ( ( a ` x ) ` ( .1. ` x ) ) ) ) e. _V
70	1, 69	eqeltri 2697	. . . 4 |- M e. _V
71	70	elixp 7915	. . 3 |- ( M e. X_ z e. ( ( O Func S ) X. B ) ( ( ( 1st ` Z ) ` z ) ( Hom ` T ) ( ( 1st ` E ) ` z ) ) <-> ( M Fn ( ( O Func S ) X. B ) /\ A. z e. ( ( O Func S ) X. B ) ( M ` z ) e. ( ( ( 1st ` Z ) ` z ) ( Hom ` T ) ( ( 1st ` E ) ` z ) ) ) )
72	5, 65, 71	sylanbrc 698	. 2 |- ( ph -> M e. X_ z e. ( ( O Func S ) X. B ) ( ( ( 1st ` Z ) ` z ) ( Hom ` T ) ( ( 1st ` E ) ` z ) ) )
73	17	adantr 481	. . . . 5 |- ( ( ph /\ ( z e. ( ( O Func S ) X. B ) /\ w e. ( ( O Func S ) X. B ) /\ g e. ( z ( Hom ` ( Q X.c O ) ) w ) ) ) -> C e. Cat )
74	19	adantr 481	. . . . 5 |- ( ( ph /\ ( z e. ( ( O Func S ) X. B ) /\ w e. ( ( O Func S ) X. B ) /\ g e. ( z ( Hom ` ( Q X.c O ) ) w ) ) ) -> V e. W )
75	21	adantr 481	. . . . 5 |- ( ( ph /\ ( z e. ( ( O Func S ) X. B ) /\ w e. ( ( O Func S ) X. B ) /\ g e. ( z ( Hom ` ( Q X.c O ) ) w ) ) ) -> ran ( Hom f ` C ) C_ U )
76	23	adantr 481	. . . . 5 |- ( ( ph /\ ( z e. ( ( O Func S ) X. B ) /\ w e. ( ( O Func S ) X. B ) /\ g e. ( z ( Hom ` ( Q X.c O ) ) w ) ) ) -> ( ran ( Hom f ` Q ) u. U ) C_ V )
77		simpr1 1067	. . . . . 6 |- ( ( ph /\ ( z e. ( ( O Func S ) X. B ) /\ w e. ( ( O Func S ) X. B ) /\ g e. ( z ( Hom ` ( Q X.c O ) ) w ) ) ) -> z e. ( ( O Func S ) X. B ) )
78		xp1st 7198	. . . . . 6 |- ( z e. ( ( O Func S ) X. B ) -> ( 1st ` z ) e. ( O Func S ) )
79	77, 78	syl 17	. . . . 5 |- ( ( ph /\ ( z e. ( ( O Func S ) X. B ) /\ w e. ( ( O Func S ) X. B ) /\ g e. ( z ( Hom ` ( Q X.c O ) ) w ) ) ) -> ( 1st ` z ) e. ( O Func S ) )
80		xp2nd 7199	. . . . . 6 |- ( z e. ( ( O Func S ) X. B ) -> ( 2nd ` z ) e. B )
81	77, 80	syl 17	. . . . 5 |- ( ( ph /\ ( z e. ( ( O Func S ) X. B ) /\ w e. ( ( O Func S ) X. B ) /\ g e. ( z ( Hom ` ( Q X.c O ) ) w ) ) ) -> ( 2nd ` z ) e. B )
82		simpr2 1068	. . . . . 6 |- ( ( ph /\ ( z e. ( ( O Func S ) X. B ) /\ w e. ( ( O Func S ) X. B ) /\ g e. ( z ( Hom ` ( Q X.c O ) ) w ) ) ) -> w e. ( ( O Func S ) X. B ) )
83		xp1st 7198	. . . . . 6 |- ( w e. ( ( O Func S ) X. B ) -> ( 1st ` w ) e. ( O Func S ) )
84	82, 83	syl 17	. . . . 5 |- ( ( ph /\ ( z e. ( ( O Func S ) X. B ) /\ w e. ( ( O Func S ) X. B ) /\ g e. ( z ( Hom ` ( Q X.c O ) ) w ) ) ) -> ( 1st ` w ) e. ( O Func S ) )
85		xp2nd 7199	. . . . . 6 |- ( w e. ( ( O Func S ) X. B ) -> ( 2nd ` w ) e. B )
86	82, 85	syl 17	. . . . 5 |- ( ( ph /\ ( z e. ( ( O Func S ) X. B ) /\ w e. ( ( O Func S ) X. B ) /\ g e. ( z ( Hom ` ( Q X.c O ) ) w ) ) ) -> ( 2nd ` w ) e. B )
87		simpr3 1069	. . . . . . 7 |- ( ( ph /\ ( z e. ( ( O Func S ) X. B ) /\ w e. ( ( O Func S ) X. B ) /\ g e. ( z ( Hom ` ( Q X.c O ) ) w ) ) ) -> g e. ( z ( Hom ` ( Q X.c O ) ) w ) )
88		eqid 2622	. . . . . . . . . 10 |- ( O Nat S ) = ( O Nat S )
89	12, 88	fuchom 16621	. . . . . . . . 9 |- ( O Nat S ) = ( Hom ` Q )
90		eqid 2622	. . . . . . . . 9 |- ( Hom ` O ) = ( Hom ` O )
91		eqid 2622	. . . . . . . . 9 |- ( Hom ` ( Q X.c O ) ) = ( Hom ` ( Q X.c O ) )
92	30, 33, 89, 90, 91, 77, 82	xpchom 16820	. . . . . . . 8 |- ( ( ph /\ ( z e. ( ( O Func S ) X. B ) /\ w e. ( ( O Func S ) X. B ) /\ g e. ( z ( Hom ` ( Q X.c O ) ) w ) ) ) -> ( z ( Hom ` ( Q X.c O ) ) w ) = ( ( ( 1st ` z ) ( O Nat S ) ( 1st ` w ) ) X. ( ( 2nd ` z ) ( Hom ` O ) ( 2nd ` w ) ) ) )
93		eqid 2622	. . . . . . . . . 10 |- ( Hom ` C ) = ( Hom ` C )
94	93, 9	oppchom 16375	. . . . . . . . 9 |- ( ( 2nd ` z ) ( Hom ` O ) ( 2nd ` w ) ) = ( ( 2nd ` w ) ( Hom ` C ) ( 2nd ` z ) )
95	94	xpeq2i 5136	. . . . . . . 8 |- ( ( ( 1st ` z ) ( O Nat S ) ( 1st ` w ) ) X. ( ( 2nd ` z ) ( Hom ` O ) ( 2nd ` w ) ) ) = ( ( ( 1st ` z ) ( O Nat S ) ( 1st ` w ) ) X. ( ( 2nd ` w ) ( Hom ` C ) ( 2nd ` z ) ) )
96	92, 95	syl6eq 2672	. . . . . . 7 |- ( ( ph /\ ( z e. ( ( O Func S ) X. B ) /\ w e. ( ( O Func S ) X. B ) /\ g e. ( z ( Hom ` ( Q X.c O ) ) w ) ) ) -> ( z ( Hom ` ( Q X.c O ) ) w ) = ( ( ( 1st ` z ) ( O Nat S ) ( 1st ` w ) ) X. ( ( 2nd ` w ) ( Hom ` C ) ( 2nd ` z ) ) ) )
97	87, 96	eleqtrd 2703	. . . . . 6 |- ( ( ph /\ ( z e. ( ( O Func S ) X. B ) /\ w e. ( ( O Func S ) X. B ) /\ g e. ( z ( Hom ` ( Q X.c O ) ) w ) ) ) -> g e. ( ( ( 1st ` z ) ( O Nat S ) ( 1st ` w ) ) X. ( ( 2nd ` w ) ( Hom ` C ) ( 2nd ` z ) ) ) )
98		xp1st 7198	. . . . . 6 |- ( g e. ( ( ( 1st ` z ) ( O Nat S ) ( 1st ` w ) ) X. ( ( 2nd ` w ) ( Hom ` C ) ( 2nd ` z ) ) ) -> ( 1st ` g ) e. ( ( 1st ` z ) ( O Nat S ) ( 1st ` w ) ) )
99	97, 98	syl 17	. . . . 5 |- ( ( ph /\ ( z e. ( ( O Func S ) X. B ) /\ w e. ( ( O Func S ) X. B ) /\ g e. ( z ( Hom ` ( Q X.c O ) ) w ) ) ) -> ( 1st ` g ) e. ( ( 1st ` z ) ( O Nat S ) ( 1st ` w ) ) )
100		xp2nd 7199	. . . . . 6 |- ( g e. ( ( ( 1st ` z ) ( O Nat S ) ( 1st ` w ) ) X. ( ( 2nd ` w ) ( Hom ` C ) ( 2nd ` z ) ) ) -> ( 2nd ` g ) e. ( ( 2nd ` w ) ( Hom ` C ) ( 2nd ` z ) ) )
101	97, 100	syl 17	. . . . 5 |- ( ( ph /\ ( z e. ( ( O Func S ) X. B ) /\ w e. ( ( O Func S ) X. B ) /\ g e. ( z ( Hom ` ( Q X.c O ) ) w ) ) ) -> ( 2nd ` g ) e. ( ( 2nd ` w ) ( Hom ` C ) ( 2nd ` z ) ) )
102	6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 73, 74, 75, 76, 79, 81, 84, 86, 99, 101, 1	yonedalem3b 16919	. . . 4 |- ( ( ph /\ ( z e. ( ( O Func S ) X. B ) /\ w e. ( ( O Func S ) X. B ) /\ g e. ( z ( Hom ` ( Q X.c O ) ) w ) ) ) -> ( ( ( 1st ` w ) M ( 2nd ` w ) ) ( <. ( ( 1st ` z ) ( 1st ` Z ) ( 2nd ` z ) ) , ( ( 1st ` w ) ( 1st ` Z ) ( 2nd ` w ) ) >. (comp ` T ) ( ( 1st ` w ) ( 1st ` E ) ( 2nd ` w ) ) ) ( ( 1st ` g ) ( <. ( 1st ` z ) , ( 2nd ` z ) >. ( 2nd ` Z ) <. ( 1st ` w ) , ( 2nd ` w ) >. ) ( 2nd ` g ) ) ) = ( ( ( 1st ` g ) ( <. ( 1st ` z ) , ( 2nd ` z ) >. ( 2nd ` E ) <. ( 1st ` w ) , ( 2nd ` w ) >. ) ( 2nd ` g ) ) ( <. ( ( 1st ` z ) ( 1st ` Z ) ( 2nd ` z ) ) , ( ( 1st ` z ) ( 1st ` E ) ( 2nd ` z ) ) >. (comp ` T ) ( ( 1st ` w ) ( 1st ` E ) ( 2nd ` w ) ) ) ( ( 1st ` z ) M ( 2nd ` z ) ) ) )
103		1st2nd2 7205	. . . . . . . . . 10 |- ( z e. ( ( O Func S ) X. B ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
104	77, 103	syl 17	. . . . . . . . 9 |- ( ( ph /\ ( z e. ( ( O Func S ) X. B ) /\ w e. ( ( O Func S ) X. B ) /\ g e. ( z ( Hom ` ( Q X.c O ) ) w ) ) ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
105	104	fveq2d 6195	. . . . . . . 8 |- ( ( ph /\ ( z e. ( ( O Func S ) X. B ) /\ w e. ( ( O Func S ) X. B ) /\ g e. ( z ( Hom ` ( Q X.c O ) ) w ) ) ) -> ( ( 1st ` Z ) ` z ) = ( ( 1st ` Z ) ` <. ( 1st ` z ) , ( 2nd ` z ) >. ) )
106		df-ov 6653	. . . . . . . 8 |- ( ( 1st ` z ) ( 1st ` Z ) ( 2nd ` z ) ) = ( ( 1st ` Z ) ` <. ( 1st ` z ) , ( 2nd ` z ) >. )
107	105, 106	syl6eqr 2674	. . . . . . 7 |- ( ( ph /\ ( z e. ( ( O Func S ) X. B ) /\ w e. ( ( O Func S ) X. B ) /\ g e. ( z ( Hom ` ( Q X.c O ) ) w ) ) ) -> ( ( 1st ` Z ) ` z ) = ( ( 1st ` z ) ( 1st ` Z ) ( 2nd ` z ) ) )
108		1st2nd2 7205	. . . . . . . . . 10 |- ( w e. ( ( O Func S ) X. B ) -> w = <. ( 1st ` w ) , ( 2nd ` w ) >. )
109	82, 108	syl 17	. . . . . . . . 9 |- ( ( ph /\ ( z e. ( ( O Func S ) X. B ) /\ w e. ( ( O Func S ) X. B ) /\ g e. ( z ( Hom ` ( Q X.c O ) ) w ) ) ) -> w = <. ( 1st ` w ) , ( 2nd ` w ) >. )
110	109	fveq2d 6195	. . . . . . . 8 |- ( ( ph /\ ( z e. ( ( O Func S ) X. B ) /\ w e. ( ( O Func S ) X. B ) /\ g e. ( z ( Hom ` ( Q X.c O ) ) w ) ) ) -> ( ( 1st ` Z ) ` w ) = ( ( 1st ` Z ) ` <. ( 1st ` w ) , ( 2nd ` w ) >. ) )
111		df-ov 6653	. . . . . . . 8 |- ( ( 1st ` w ) ( 1st ` Z ) ( 2nd ` w ) ) = ( ( 1st ` Z ) ` <. ( 1st ` w ) , ( 2nd ` w ) >. )
112	110, 111	syl6eqr 2674	. . . . . . 7 |- ( ( ph /\ ( z e. ( ( O Func S ) X. B ) /\ w e. ( ( O Func S ) X. B ) /\ g e. ( z ( Hom ` ( Q X.c O ) ) w ) ) ) -> ( ( 1st ` Z ) ` w ) = ( ( 1st ` w ) ( 1st ` Z ) ( 2nd ` w ) ) )
113	107, 112	opeq12d 4410	. . . . . 6 |- ( ( ph /\ ( z e. ( ( O Func S ) X. B ) /\ w e. ( ( O Func S ) X. B ) /\ g e. ( z ( Hom ` ( Q X.c O ) ) w ) ) ) -> <. ( ( 1st ` Z ) ` z ) , ( ( 1st ` Z ) ` w ) >. = <. ( ( 1st ` z ) ( 1st ` Z ) ( 2nd ` z ) ) , ( ( 1st ` w ) ( 1st ` Z ) ( 2nd ` w ) ) >. )
114	109	fveq2d 6195	. . . . . . 7 |- ( ( ph /\ ( z e. ( ( O Func S ) X. B ) /\ w e. ( ( O Func S ) X. B ) /\ g e. ( z ( Hom ` ( Q X.c O ) ) w ) ) ) -> ( ( 1st ` E ) ` w ) = ( ( 1st ` E ) ` <. ( 1st ` w ) , ( 2nd ` w ) >. ) )
115		df-ov 6653	. . . . . . 7 |- ( ( 1st ` w ) ( 1st ` E ) ( 2nd ` w ) ) = ( ( 1st ` E ) ` <. ( 1st ` w ) , ( 2nd ` w ) >. )
116	114, 115	syl6eqr 2674	. . . . . 6 |- ( ( ph /\ ( z e. ( ( O Func S ) X. B ) /\ w e. ( ( O Func S ) X. B ) /\ g e. ( z ( Hom ` ( Q X.c O ) ) w ) ) ) -> ( ( 1st ` E ) ` w ) = ( ( 1st ` w ) ( 1st ` E ) ( 2nd ` w ) ) )
117	113, 116	oveq12d 6668	. . . . 5 |- ( ( ph /\ ( z e. ( ( O Func S ) X. B ) /\ w e. ( ( O Func S ) X. B ) /\ g e. ( z ( Hom ` ( Q X.c O ) ) w ) ) ) -> ( <. ( ( 1st ` Z ) ` z ) , ( ( 1st ` Z ) ` w ) >. (comp ` T ) ( ( 1st ` E ) ` w ) ) = ( <. ( ( 1st ` z ) ( 1st ` Z ) ( 2nd ` z ) ) , ( ( 1st ` w ) ( 1st ` Z ) ( 2nd ` w ) ) >. (comp ` T ) ( ( 1st ` w ) ( 1st ` E ) ( 2nd ` w ) ) ) )
118	109	fveq2d 6195	. . . . . 6 |- ( ( ph /\ ( z e. ( ( O Func S ) X. B ) /\ w e. ( ( O Func S ) X. B ) /\ g e. ( z ( Hom ` ( Q X.c O ) ) w ) ) ) -> ( M ` w ) = ( M ` <. ( 1st ` w ) , ( 2nd ` w ) >. ) )
119		df-ov 6653	. . . . . 6 |- ( ( 1st ` w ) M ( 2nd ` w ) ) = ( M ` <. ( 1st ` w ) , ( 2nd ` w ) >. )
120	118, 119	syl6eqr 2674	. . . . 5 |- ( ( ph /\ ( z e. ( ( O Func S ) X. B ) /\ w e. ( ( O Func S ) X. B ) /\ g e. ( z ( Hom ` ( Q X.c O ) ) w ) ) ) -> ( M ` w ) = ( ( 1st ` w ) M ( 2nd ` w ) ) )
121	104, 109	oveq12d 6668	. . . . . . 7 |- ( ( ph /\ ( z e. ( ( O Func S ) X. B ) /\ w e. ( ( O Func S ) X. B ) /\ g e. ( z ( Hom ` ( Q X.c O ) ) w ) ) ) -> ( z ( 2nd ` Z ) w ) = ( <. ( 1st ` z ) , ( 2nd ` z ) >. ( 2nd ` Z ) <. ( 1st ` w ) , ( 2nd ` w ) >. ) )
122		1st2nd2 7205	. . . . . . . 8 |- ( g e. ( ( ( 1st ` z ) ( O Nat S ) ( 1st ` w ) ) X. ( ( 2nd ` w ) ( Hom ` C ) ( 2nd ` z ) ) ) -> g = <. ( 1st ` g ) , ( 2nd ` g ) >. )
123	97, 122	syl 17	. . . . . . 7 |- ( ( ph /\ ( z e. ( ( O Func S ) X. B ) /\ w e. ( ( O Func S ) X. B ) /\ g e. ( z ( Hom ` ( Q X.c O ) ) w ) ) ) -> g = <. ( 1st ` g ) , ( 2nd ` g ) >. )
124	121, 123	fveq12d 6197	. . . . . 6 |- ( ( ph /\ ( z e. ( ( O Func S ) X. B ) /\ w e. ( ( O Func S ) X. B ) /\ g e. ( z ( Hom ` ( Q X.c O ) ) w ) ) ) -> ( ( z ( 2nd ` Z ) w ) ` g ) = ( ( <. ( 1st ` z ) , ( 2nd ` z ) >. ( 2nd ` Z ) <. ( 1st ` w ) , ( 2nd ` w ) >. ) ` <. ( 1st ` g ) , ( 2nd ` g ) >. ) )
125		df-ov 6653	. . . . . 6 |- ( ( 1st ` g ) ( <. ( 1st ` z ) , ( 2nd ` z ) >. ( 2nd ` Z ) <. ( 1st ` w ) , ( 2nd ` w ) >. ) ( 2nd ` g ) ) = ( ( <. ( 1st ` z ) , ( 2nd ` z ) >. ( 2nd ` Z ) <. ( 1st ` w ) , ( 2nd ` w ) >. ) ` <. ( 1st ` g ) , ( 2nd ` g ) >. )
126	124, 125	syl6eqr 2674	. . . . 5 |- ( ( ph /\ ( z e. ( ( O Func S ) X. B ) /\ w e. ( ( O Func S ) X. B ) /\ g e. ( z ( Hom ` ( Q X.c O ) ) w ) ) ) -> ( ( z ( 2nd ` Z ) w ) ` g ) = ( ( 1st ` g ) ( <. ( 1st ` z ) , ( 2nd ` z ) >. ( 2nd ` Z ) <. ( 1st ` w ) , ( 2nd ` w ) >. ) ( 2nd ` g ) ) )
127	117, 120, 126	oveq123d 6671	. . . 4 |- ( ( ph /\ ( z e. ( ( O Func S ) X. B ) /\ w e. ( ( O Func S ) X. B ) /\ g e. ( z ( Hom ` ( Q X.c O ) ) w ) ) ) -> ( ( M ` w ) ( <. ( ( 1st ` Z ) ` z ) , ( ( 1st ` Z ) ` w ) >. (comp ` T ) ( ( 1st ` E ) ` w ) ) ( ( z ( 2nd ` Z ) w ) ` g ) ) = ( ( ( 1st ` w ) M ( 2nd ` w ) ) ( <. ( ( 1st ` z ) ( 1st ` Z ) ( 2nd ` z ) ) , ( ( 1st ` w ) ( 1st ` Z ) ( 2nd ` w ) ) >. (comp ` T ) ( ( 1st ` w ) ( 1st ` E ) ( 2nd ` w ) ) ) ( ( 1st ` g ) ( <. ( 1st ` z ) , ( 2nd ` z ) >. ( 2nd ` Z ) <. ( 1st ` w ) , ( 2nd ` w ) >. ) ( 2nd ` g ) ) ) )
128	104	fveq2d 6195	. . . . . . . 8 |- ( ( ph /\ ( z e. ( ( O Func S ) X. B ) /\ w e. ( ( O Func S ) X. B ) /\ g e. ( z ( Hom ` ( Q X.c O ) ) w ) ) ) -> ( ( 1st ` E ) ` z ) = ( ( 1st ` E ) ` <. ( 1st ` z ) , ( 2nd ` z ) >. ) )
129		df-ov 6653	. . . . . . . 8 |- ( ( 1st ` z ) ( 1st ` E ) ( 2nd ` z ) ) = ( ( 1st ` E ) ` <. ( 1st ` z ) , ( 2nd ` z ) >. )
130	128, 129	syl6eqr 2674	. . . . . . 7 |- ( ( ph /\ ( z e. ( ( O Func S ) X. B ) /\ w e. ( ( O Func S ) X. B ) /\ g e. ( z ( Hom ` ( Q X.c O ) ) w ) ) ) -> ( ( 1st ` E ) ` z ) = ( ( 1st ` z ) ( 1st ` E ) ( 2nd ` z ) ) )
131	107, 130	opeq12d 4410	. . . . . 6 |- ( ( ph /\ ( z e. ( ( O Func S ) X. B ) /\ w e. ( ( O Func S ) X. B ) /\ g e. ( z ( Hom ` ( Q X.c O ) ) w ) ) ) -> <. ( ( 1st ` Z ) ` z ) , ( ( 1st ` E ) ` z ) >. = <. ( ( 1st ` z ) ( 1st ` Z ) ( 2nd ` z ) ) , ( ( 1st ` z ) ( 1st ` E ) ( 2nd ` z ) ) >. )
132	131, 116	oveq12d 6668	. . . . 5 |- ( ( ph /\ ( z e. ( ( O Func S ) X. B ) /\ w e. ( ( O Func S ) X. B ) /\ g e. ( z ( Hom ` ( Q X.c O ) ) w ) ) ) -> ( <. ( ( 1st ` Z ) ` z ) , ( ( 1st ` E ) ` z ) >. (comp ` T ) ( ( 1st ` E ) ` w ) ) = ( <. ( ( 1st ` z ) ( 1st ` Z ) ( 2nd ` z ) ) , ( ( 1st ` z ) ( 1st ` E ) ( 2nd ` z ) ) >. (comp ` T ) ( ( 1st ` w ) ( 1st ` E ) ( 2nd ` w ) ) ) )
133	104, 109	oveq12d 6668	. . . . . . 7 |- ( ( ph /\ ( z e. ( ( O Func S ) X. B ) /\ w e. ( ( O Func S ) X. B ) /\ g e. ( z ( Hom ` ( Q X.c O ) ) w ) ) ) -> ( z ( 2nd ` E ) w ) = ( <. ( 1st ` z ) , ( 2nd ` z ) >. ( 2nd ` E ) <. ( 1st ` w ) , ( 2nd ` w ) >. ) )
134	133, 123	fveq12d 6197	. . . . . 6 |- ( ( ph /\ ( z e. ( ( O Func S ) X. B ) /\ w e. ( ( O Func S ) X. B ) /\ g e. ( z ( Hom ` ( Q X.c O ) ) w ) ) ) -> ( ( z ( 2nd ` E ) w ) ` g ) = ( ( <. ( 1st ` z ) , ( 2nd ` z ) >. ( 2nd ` E ) <. ( 1st ` w ) , ( 2nd ` w ) >. ) ` <. ( 1st ` g ) , ( 2nd ` g ) >. ) )
135		df-ov 6653	. . . . . 6 |- ( ( 1st ` g ) ( <. ( 1st ` z ) , ( 2nd ` z ) >. ( 2nd ` E ) <. ( 1st ` w ) , ( 2nd ` w ) >. ) ( 2nd ` g ) ) = ( ( <. ( 1st ` z ) , ( 2nd ` z ) >. ( 2nd ` E ) <. ( 1st ` w ) , ( 2nd ` w ) >. ) ` <. ( 1st ` g ) , ( 2nd ` g ) >. )
136	134, 135	syl6eqr 2674	. . . . 5 |- ( ( ph /\ ( z e. ( ( O Func S ) X. B ) /\ w e. ( ( O Func S ) X. B ) /\ g e. ( z ( Hom ` ( Q X.c O ) ) w ) ) ) -> ( ( z ( 2nd ` E ) w ) ` g ) = ( ( 1st ` g ) ( <. ( 1st ` z ) , ( 2nd ` z ) >. ( 2nd ` E ) <. ( 1st ` w ) , ( 2nd ` w ) >. ) ( 2nd ` g ) ) )
137	104	fveq2d 6195	. . . . . 6 |- ( ( ph /\ ( z e. ( ( O Func S ) X. B ) /\ w e. ( ( O Func S ) X. B ) /\ g e. ( z ( Hom ` ( Q X.c O ) ) w ) ) ) -> ( M ` z ) = ( M ` <. ( 1st ` z ) , ( 2nd ` z ) >. ) )
138		df-ov 6653	. . . . . 6 |- ( ( 1st ` z ) M ( 2nd ` z ) ) = ( M ` <. ( 1st ` z ) , ( 2nd ` z ) >. )
139	137, 138	syl6eqr 2674	. . . . 5 |- ( ( ph /\ ( z e. ( ( O Func S ) X. B ) /\ w e. ( ( O Func S ) X. B ) /\ g e. ( z ( Hom ` ( Q X.c O ) ) w ) ) ) -> ( M ` z ) = ( ( 1st ` z ) M ( 2nd ` z ) ) )
140	132, 136, 139	oveq123d 6671	. . . 4 |- ( ( ph /\ ( z e. ( ( O Func S ) X. B ) /\ w e. ( ( O Func S ) X. B ) /\ g e. ( z ( Hom ` ( Q X.c O ) ) w ) ) ) -> ( ( ( z ( 2nd ` E ) w ) ` g ) ( <. ( ( 1st ` Z ) ` z ) , ( ( 1st ` E ) ` z ) >. (comp ` T ) ( ( 1st ` E ) ` w ) ) ( M ` z ) ) = ( ( ( 1st ` g ) ( <. ( 1st ` z ) , ( 2nd ` z ) >. ( 2nd ` E ) <. ( 1st ` w ) , ( 2nd ` w ) >. ) ( 2nd ` g ) ) ( <. ( ( 1st ` z ) ( 1st ` Z ) ( 2nd ` z ) ) , ( ( 1st ` z ) ( 1st ` E ) ( 2nd ` z ) ) >. (comp ` T ) ( ( 1st ` w ) ( 1st ` E ) ( 2nd ` w ) ) ) ( ( 1st ` z ) M ( 2nd ` z ) ) ) )
141	102, 127, 140	3eqtr4d 2666	. . 3 |- ( ( ph /\ ( z e. ( ( O Func S ) X. B ) /\ w e. ( ( O Func S ) X. B ) /\ g e. ( z ( Hom ` ( Q X.c O ) ) w ) ) ) -> ( ( M ` w ) ( <. ( ( 1st ` Z ) ` z ) , ( ( 1st ` Z ) ` w ) >. (comp ` T ) ( ( 1st ` E ) ` w ) ) ( ( z ( 2nd ` Z ) w ) ` g ) ) = ( ( ( z ( 2nd ` E ) w ) ` g ) ( <. ( ( 1st ` Z ) ` z ) , ( ( 1st ` E ) ` z ) >. (comp ` T ) ( ( 1st ` E ) ` w ) ) ( M ` z ) ) )
142	141	ralrimivvva 2972	. 2 |- ( ph -> A. z e. ( ( O Func S ) X. B ) A. w e. ( ( O Func S ) X. B ) A. g e. ( z ( Hom ` ( Q X.c O ) ) w ) ( ( M ` w ) ( <. ( ( 1st ` Z ) ` z ) , ( ( 1st ` Z ) ` w ) >. (comp ` T ) ( ( 1st ` E ) ` w ) ) ( ( z ( 2nd ` Z ) w ) ` g ) ) = ( ( ( z ( 2nd ` E ) w ) ` g ) ( <. ( ( 1st ` Z ) ` z ) , ( ( 1st ` E ) ` z ) >. (comp ` T ) ( ( 1st ` E ) ` w ) ) ( M ` z ) ) )
143		eqid 2622	. . 3 |- ( ( Q X.c O ) Nat T ) = ( ( Q X.c O ) Nat T )
144		eqid 2622	. . 3 |- (comp ` T ) = (comp ` T )
145	143, 33, 91, 29, 144, 37, 44	isnat2 16608	. 2 |- ( ph -> ( M e. ( Z ( ( Q X.c O ) Nat T ) E ) <-> ( M e. X_ z e. ( ( O Func S ) X. B ) ( ( ( 1st ` Z ) ` z ) ( Hom ` T ) ( ( 1st ` E ) ` z ) ) /\ A. z e. ( ( O Func S ) X. B ) A. w e. ( ( O Func S ) X. B ) A. g e. ( z ( Hom ` ( Q X.c O ) ) w ) ( ( M ` w ) ( <. ( ( 1st ` Z ) ` z ) , ( ( 1st ` Z ) ` w ) >. (comp ` T ) ( ( 1st ` E ) ` w ) ) ( ( z ( 2nd ` Z ) w ) ` g ) ) = ( ( ( z ( 2nd ` E ) w ) ` g ) ( <. ( ( 1st ` Z ) ` z ) , ( ( 1st ` E ) ` z ) >. (comp ` T ) ( ( 1st ` E ) ` w ) ) ( M ` z ) ) ) ) )
146	72, 142, 145	mpbir2and 957	1 |- ( ph -> M e. ( Z ( ( Q X.c O ) Nat T ) E ) )

Syntax hints:   -> wi 4   /\ 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   X. cxp 5112   ran crn 5115   Rel wrel 5119   Fn wfn 5883   -->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-pm 7860  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-ress 15865  df-hom 15966  df-cco 15967  df-cat 16329  df-cid 16330  df-homf 16331  df-comf 16332  df-oppc 16372  df-ssc 16470  df-resc 16471  df-subc 16472  df-func 16518  df-cofu 16520  df-nat 16603  df-fuc 16604  df-setc 16726  df-xpc 16812  df-1stf 16813  df-2ndf 16814  df-prf 16815  df-evlf 16853  df-curf 16854  df-hof 16890  df-yon 16891

This theorem is referenced by:  yonedainv  16921