Theorem yonedalem22 16918

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 )
yonedalem22.g	|- ( ph -> G e. ( O Func S ) )
yonedalem22.p	|- ( ph -> P e. B )
yonedalem22.a	|- ( ph -> A e. ( F ( O Nat S ) G ) )
yonedalem22.k	|- ( ph -> K e. ( P ( Hom ` C ) X ) )

Assertion

Ref	Expression
yonedalem22	|- ( ph -> ( A ( <. F , X >. ( 2nd ` Z ) <. G , P >. ) K ) = ( ( ( P ( 2nd ` Y ) X ) ` K ) ( <. ( ( 1st ` Y ) ` X ) , F >. ( 2nd ` H ) <. ( ( 1st ` Y ) ` P ) , G >. ) A ) )

Proof of Theorem yonedalem22

Step	Hyp	Ref	Expression
1		yoneda.z	. . . . . . 7 |- Z = ( H o.func ( ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ⟨,⟩F ( Q 1stF O ) ) )
2	1	fveq2i 6194	. . . . . 6 |- ( 2nd ` Z ) = ( 2nd ` ( H o.func ( ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ⟨,⟩F ( Q 1stF O ) ) ) )
3	2	oveqi 6663	. . . . 5 |- ( <. F , X >. ( 2nd ` Z ) <. G , P >. ) = ( <. F , X >. ( 2nd ` ( H o.func ( ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ⟨,⟩F ( Q 1stF O ) ) ) ) <. G , P >. )
4	3	oveqi 6663	. . . 4 |- ( A ( <. F , X >. ( 2nd ` Z ) <. G , P >. ) K ) = ( A ( <. F , X >. ( 2nd ` ( H o.func ( ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ⟨,⟩F ( Q 1stF O ) ) ) ) <. G , P >. ) K )
5		df-ov 6653	. . . 4 |- ( A ( <. F , X >. ( 2nd ` ( H o.func ( ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ⟨,⟩F ( Q 1stF O ) ) ) ) <. G , P >. ) K ) = ( ( <. F , X >. ( 2nd ` ( H o.func ( ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ⟨,⟩F ( Q 1stF O ) ) ) ) <. G , P >. ) ` <. A , K >. )
6	4, 5	eqtri 2644	. . 3 |- ( A ( <. F , X >. ( 2nd ` Z ) <. G , P >. ) K ) = ( ( <. F , X >. ( 2nd ` ( H o.func ( ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ⟨,⟩F ( Q 1stF O ) ) ) ) <. G , P >. ) ` <. A , K >. )
7		eqid 2622	. . . . 5 |- ( Q X.c O ) = ( Q X.c O )
8		yoneda.q	. . . . . 6 |- Q = ( O FuncCat S )
9	8	fucbas 16620	. . . . 5 |- ( O Func S ) = ( Base ` Q )
10		yoneda.o	. . . . . 6 |- O = (oppCat ` C )
11		yoneda.b	. . . . . 6 |- B = ( Base ` C )
12	10, 11	oppcbas 16378	. . . . 5 |- B = ( Base ` O )
13	7, 9, 12	xpcbas 16818	. . . 4 |- ( ( O Func S ) X. B ) = ( Base ` ( Q X.c O ) )
14		eqid 2622	. . . . 5 |- ( ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ⟨,⟩F ( Q 1stF O ) ) = ( ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ⟨,⟩F ( Q 1stF O ) )
15		eqid 2622	. . . . 5 |- ( (oppCat ` Q ) X.c Q ) = ( (oppCat ` Q ) X.c Q )
16		yoneda.c	. . . . . . . . 9 |- ( ph -> C e. Cat )
17	10	oppccat 16382	. . . . . . . . 9 |- ( C e. Cat -> O e. Cat )
18	16, 17	syl 17	. . . . . . . 8 |- ( ph -> O e. Cat )
19		yoneda.w	. . . . . . . . . 10 |- ( ph -> V e. W )
20		yoneda.v	. . . . . . . . . . 11 |- ( ph -> ( ran ( Hom f ` Q ) u. U ) C_ V )
21	20	unssbd 3791	. . . . . . . . . 10 |- ( ph -> U C_ V )
22	19, 21	ssexd 4805	. . . . . . . . 9 |- ( ph -> U e. _V )
23		yoneda.s	. . . . . . . . . 10 |- S = ( SetCat ` U )
24	23	setccat 16735	. . . . . . . . 9 |- ( U e. _V -> S e. Cat )
25	22, 24	syl 17	. . . . . . . 8 |- ( ph -> S e. Cat )
26	8, 18, 25	fuccat 16630	. . . . . . 7 |- ( ph -> Q e. Cat )
27		eqid 2622	. . . . . . 7 |- ( Q 2ndF O ) = ( Q 2ndF O )
28	7, 26, 18, 27	2ndfcl 16838	. . . . . 6 |- ( ph -> ( Q 2ndF O ) e. ( ( Q X.c O ) Func O ) )
29		eqid 2622	. . . . . . . 8 |- (oppCat ` Q ) = (oppCat ` Q )
30		relfunc 16522	. . . . . . . . 9 |- Rel ( C Func Q )
31		yoneda.y	. . . . . . . . . 10 |- Y = (Yon ` C )
32		yoneda.u	. . . . . . . . . 10 |- ( ph -> ran ( Hom f ` C ) C_ U )
33	31, 16, 10, 23, 8, 22, 32	yoncl 16902	. . . . . . . . 9 |- ( ph -> Y e. ( C Func Q ) )
34		1st2ndbr 7217	. . . . . . . . 9 |- ( ( Rel ( C Func Q ) /\ Y e. ( C Func Q ) ) -> ( 1st ` Y ) ( C Func Q ) ( 2nd ` Y ) )
35	30, 33, 34	sylancr 695	. . . . . . . 8 |- ( ph -> ( 1st ` Y ) ( C Func Q ) ( 2nd ` Y ) )
36	10, 29, 35	funcoppc 16535	. . . . . . 7 |- ( ph -> ( 1st ` Y ) ( O Func (oppCat ` Q ) )tpos ( 2nd ` Y ) )
37		df-br 4654	. . . . . . 7 |- ( ( 1st ` Y ) ( O Func (oppCat ` Q ) )tpos ( 2nd ` Y ) <-> <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. e. ( O Func (oppCat ` Q ) ) )
38	36, 37	sylib 208	. . . . . 6 |- ( ph -> <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. e. ( O Func (oppCat ` Q ) ) )
39	28, 38	cofucl 16548	. . . . 5 |- ( ph -> ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) e. ( ( Q X.c O ) Func (oppCat ` Q ) ) )
40		eqid 2622	. . . . . 6 |- ( Q 1stF O ) = ( Q 1stF O )
41	7, 26, 18, 40	1stfcl 16837	. . . . 5 |- ( ph -> ( Q 1stF O ) e. ( ( Q X.c O ) Func Q ) )
42	14, 15, 39, 41	prfcl 16843	. . . 4 |- ( ph -> ( ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ⟨,⟩F ( Q 1stF O ) ) e. ( ( Q X.c O ) Func ( (oppCat ` Q ) X.c Q ) ) )
43		yoneda.h	. . . . 5 |- H = (HomF ` Q )
44		yoneda.t	. . . . 5 |- T = ( SetCat ` V )
45	20	unssad 3790	. . . . 5 |- ( ph -> ran ( Hom f ` Q ) C_ V )
46	43, 29, 44, 26, 19, 45	hofcl 16899	. . . 4 |- ( ph -> H e. ( ( (oppCat ` Q ) X.c Q ) Func T ) )
47		yonedalem21.f	. . . . 5 |- ( ph -> F e. ( O Func S ) )
48		yonedalem21.x	. . . . 5 |- ( ph -> X e. B )
49		opelxpi 5148	. . . . 5 |- ( ( F e. ( O Func S ) /\ X e. B ) -> <. F , X >. e. ( ( O Func S ) X. B ) )
50	47, 48, 49	syl2anc 693	. . . 4 |- ( ph -> <. F , X >. e. ( ( O Func S ) X. B ) )
51		yonedalem22.g	. . . . 5 |- ( ph -> G e. ( O Func S ) )
52		yonedalem22.p	. . . . 5 |- ( ph -> P e. B )
53		opelxpi 5148	. . . . 5 |- ( ( G e. ( O Func S ) /\ P e. B ) -> <. G , P >. e. ( ( O Func S ) X. B ) )
54	51, 52, 53	syl2anc 693	. . . 4 |- ( ph -> <. G , P >. e. ( ( O Func S ) X. B ) )
55		eqid 2622	. . . 4 |- ( Hom ` ( Q X.c O ) ) = ( Hom ` ( Q X.c O ) )
56		yonedalem22.a	. . . . . 6 |- ( ph -> A e. ( F ( O Nat S ) G ) )
57		yonedalem22.k	. . . . . . 7 |- ( ph -> K e. ( P ( Hom ` C ) X ) )
58		eqid 2622	. . . . . . . 8 |- ( Hom ` C ) = ( Hom ` C )
59	58, 10	oppchom 16375	. . . . . . 7 |- ( X ( Hom ` O ) P ) = ( P ( Hom ` C ) X )
60	57, 59	syl6eleqr 2712	. . . . . 6 |- ( ph -> K e. ( X ( Hom ` O ) P ) )
61		opelxpi 5148	. . . . . 6 |- ( ( A e. ( F ( O Nat S ) G ) /\ K e. ( X ( Hom ` O ) P ) ) -> <. A , K >. e. ( ( F ( O Nat S ) G ) X. ( X ( Hom ` O ) P ) ) )
62	56, 60, 61	syl2anc 693	. . . . 5 |- ( ph -> <. A , K >. e. ( ( F ( O Nat S ) G ) X. ( X ( Hom ` O ) P ) ) )
63		eqid 2622	. . . . . . 7 |- ( O Nat S ) = ( O Nat S )
64	8, 63	fuchom 16621	. . . . . 6 |- ( O Nat S ) = ( Hom ` Q )
65		eqid 2622	. . . . . 6 |- ( Hom ` O ) = ( Hom ` O )
66	7, 9, 12, 64, 65, 47, 48, 51, 52, 55	xpchom2 16826	. . . . 5 |- ( ph -> ( <. F , X >. ( Hom ` ( Q X.c O ) ) <. G , P >. ) = ( ( F ( O Nat S ) G ) X. ( X ( Hom ` O ) P ) ) )
67	62, 66	eleqtrrd 2704	. . . 4 |- ( ph -> <. A , K >. e. ( <. F , X >. ( Hom ` ( Q X.c O ) ) <. G , P >. ) )
68	13, 42, 46, 50, 54, 55, 67	cofu2 16546	. . 3 |- ( ph -> ( ( <. F , X >. ( 2nd ` ( H o.func ( ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ⟨,⟩F ( Q 1stF O ) ) ) ) <. G , P >. ) ` <. A , K >. ) = ( ( ( ( 1st ` ( ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ⟨,⟩F ( Q 1stF O ) ) ) ` <. F , X >. ) ( 2nd ` H ) ( ( 1st ` ( ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ⟨,⟩F ( Q 1stF O ) ) ) ` <. G , P >. ) ) ` ( ( <. F , X >. ( 2nd ` ( ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ⟨,⟩F ( Q 1stF O ) ) ) <. G , P >. ) ` <. A , K >. ) ) )
69	6, 68	syl5eq 2668	. 2 |- ( ph -> ( A ( <. F , X >. ( 2nd ` Z ) <. G , P >. ) K ) = ( ( ( ( 1st ` ( ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ⟨,⟩F ( Q 1stF O ) ) ) ` <. F , X >. ) ( 2nd ` H ) ( ( 1st ` ( ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ⟨,⟩F ( Q 1stF O ) ) ) ` <. G , P >. ) ) ` ( ( <. F , X >. ( 2nd ` ( ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ⟨,⟩F ( Q 1stF O ) ) ) <. G , P >. ) ` <. A , K >. ) ) )
70	14, 13, 55, 39, 41, 50	prf1 16840	. . . . . 6 |- ( ph -> ( ( 1st ` ( ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ⟨,⟩F ( Q 1stF O ) ) ) ` <. F , X >. ) = <. ( ( 1st ` ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ) ` <. F , X >. ) , ( ( 1st ` ( Q 1stF O ) ) ` <. F , X >. ) >. )
71	13, 28, 38, 50	cofu1 16544	. . . . . . . 8 |- ( ph -> ( ( 1st ` ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ) ` <. F , X >. ) = ( ( 1st ` <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. ) ` ( ( 1st ` ( Q 2ndF O ) ) ` <. F , X >. ) ) )
72		fvex 6201	. . . . . . . . . . 11 |- ( 1st ` Y ) e. _V
73		fvex 6201	. . . . . . . . . . . 12 |- ( 2nd ` Y ) e. _V
74	73	tposex 7386	. . . . . . . . . . 11 |- tpos ( 2nd ` Y ) e. _V
75	72, 74	op1st 7176	. . . . . . . . . 10 |- ( 1st ` <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. ) = ( 1st ` Y )
76	75	a1i 11	. . . . . . . . 9 |- ( ph -> ( 1st ` <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. ) = ( 1st ` Y ) )
77	7, 13, 55, 26, 18, 27, 50	2ndf1 16835	. . . . . . . . . 10 |- ( ph -> ( ( 1st ` ( Q 2ndF O ) ) ` <. F , X >. ) = ( 2nd ` <. F , X >. ) )
78		op2ndg 7181	. . . . . . . . . . 11 |- ( ( F e. ( O Func S ) /\ X e. B ) -> ( 2nd ` <. F , X >. ) = X )
79	47, 48, 78	syl2anc 693	. . . . . . . . . 10 |- ( ph -> ( 2nd ` <. F , X >. ) = X )
80	77, 79	eqtrd 2656	. . . . . . . . 9 |- ( ph -> ( ( 1st ` ( Q 2ndF O ) ) ` <. F , X >. ) = X )
81	76, 80	fveq12d 6197	. . . . . . . 8 |- ( ph -> ( ( 1st ` <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. ) ` ( ( 1st ` ( Q 2ndF O ) ) ` <. F , X >. ) ) = ( ( 1st ` Y ) ` X ) )
82	71, 81	eqtrd 2656	. . . . . . 7 |- ( ph -> ( ( 1st ` ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ) ` <. F , X >. ) = ( ( 1st ` Y ) ` X ) )
83	7, 13, 55, 26, 18, 40, 50	1stf1 16832	. . . . . . . 8 |- ( ph -> ( ( 1st ` ( Q 1stF O ) ) ` <. F , X >. ) = ( 1st ` <. F , X >. ) )
84		op1stg 7180	. . . . . . . . 9 |- ( ( F e. ( O Func S ) /\ X e. B ) -> ( 1st ` <. F , X >. ) = F )
85	47, 48, 84	syl2anc 693	. . . . . . . 8 |- ( ph -> ( 1st ` <. F , X >. ) = F )
86	83, 85	eqtrd 2656	. . . . . . 7 |- ( ph -> ( ( 1st ` ( Q 1stF O ) ) ` <. F , X >. ) = F )
87	82, 86	opeq12d 4410	. . . . . 6 |- ( ph -> <. ( ( 1st ` ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ) ` <. F , X >. ) , ( ( 1st ` ( Q 1stF O ) ) ` <. F , X >. ) >. = <. ( ( 1st ` Y ) ` X ) , F >. )
88	70, 87	eqtrd 2656	. . . . 5 |- ( ph -> ( ( 1st ` ( ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ⟨,⟩F ( Q 1stF O ) ) ) ` <. F , X >. ) = <. ( ( 1st ` Y ) ` X ) , F >. )
89	14, 13, 55, 39, 41, 54	prf1 16840	. . . . . 6 |- ( ph -> ( ( 1st ` ( ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ⟨,⟩F ( Q 1stF O ) ) ) ` <. G , P >. ) = <. ( ( 1st ` ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ) ` <. G , P >. ) , ( ( 1st ` ( Q 1stF O ) ) ` <. G , P >. ) >. )
90	13, 28, 38, 54	cofu1 16544	. . . . . . . 8 |- ( ph -> ( ( 1st ` ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ) ` <. G , P >. ) = ( ( 1st ` <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. ) ` ( ( 1st ` ( Q 2ndF O ) ) ` <. G , P >. ) ) )
91	7, 13, 55, 26, 18, 27, 54	2ndf1 16835	. . . . . . . . . 10 |- ( ph -> ( ( 1st ` ( Q 2ndF O ) ) ` <. G , P >. ) = ( 2nd ` <. G , P >. ) )
92		op2ndg 7181	. . . . . . . . . . 11 |- ( ( G e. ( O Func S ) /\ P e. B ) -> ( 2nd ` <. G , P >. ) = P )
93	51, 52, 92	syl2anc 693	. . . . . . . . . 10 |- ( ph -> ( 2nd ` <. G , P >. ) = P )
94	91, 93	eqtrd 2656	. . . . . . . . 9 |- ( ph -> ( ( 1st ` ( Q 2ndF O ) ) ` <. G , P >. ) = P )
95	76, 94	fveq12d 6197	. . . . . . . 8 |- ( ph -> ( ( 1st ` <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. ) ` ( ( 1st ` ( Q 2ndF O ) ) ` <. G , P >. ) ) = ( ( 1st ` Y ) ` P ) )
96	90, 95	eqtrd 2656	. . . . . . 7 |- ( ph -> ( ( 1st ` ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ) ` <. G , P >. ) = ( ( 1st ` Y ) ` P ) )
97	7, 13, 55, 26, 18, 40, 54	1stf1 16832	. . . . . . . 8 |- ( ph -> ( ( 1st ` ( Q 1stF O ) ) ` <. G , P >. ) = ( 1st ` <. G , P >. ) )
98		op1stg 7180	. . . . . . . . 9 |- ( ( G e. ( O Func S ) /\ P e. B ) -> ( 1st ` <. G , P >. ) = G )
99	51, 52, 98	syl2anc 693	. . . . . . . 8 |- ( ph -> ( 1st ` <. G , P >. ) = G )
100	97, 99	eqtrd 2656	. . . . . . 7 |- ( ph -> ( ( 1st ` ( Q 1stF O ) ) ` <. G , P >. ) = G )
101	96, 100	opeq12d 4410	. . . . . 6 |- ( ph -> <. ( ( 1st ` ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ) ` <. G , P >. ) , ( ( 1st ` ( Q 1stF O ) ) ` <. G , P >. ) >. = <. ( ( 1st ` Y ) ` P ) , G >. )
102	89, 101	eqtrd 2656	. . . . 5 |- ( ph -> ( ( 1st ` ( ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ⟨,⟩F ( Q 1stF O ) ) ) ` <. G , P >. ) = <. ( ( 1st ` Y ) ` P ) , G >. )
103	88, 102	oveq12d 6668	. . . 4 |- ( ph -> ( ( ( 1st ` ( ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ⟨,⟩F ( Q 1stF O ) ) ) ` <. F , X >. ) ( 2nd ` H ) ( ( 1st ` ( ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ⟨,⟩F ( Q 1stF O ) ) ) ` <. G , P >. ) ) = ( <. ( ( 1st ` Y ) ` X ) , F >. ( 2nd ` H ) <. ( ( 1st ` Y ) ` P ) , G >. ) )
104	14, 13, 55, 39, 41, 50, 54, 67	prf2 16842	. . . . 5 |- ( ph -> ( ( <. F , X >. ( 2nd ` ( ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ⟨,⟩F ( Q 1stF O ) ) ) <. G , P >. ) ` <. A , K >. ) = <. ( ( <. F , X >. ( 2nd ` ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ) <. G , P >. ) ` <. A , K >. ) , ( ( <. F , X >. ( 2nd ` ( Q 1stF O ) ) <. G , P >. ) ` <. A , K >. ) >. )
105	13, 28, 38, 50, 54, 55, 67	cofu2 16546	. . . . . . 7 |- ( ph -> ( ( <. F , X >. ( 2nd ` ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ) <. G , P >. ) ` <. A , K >. ) = ( ( ( ( 1st ` ( Q 2ndF O ) ) ` <. F , X >. ) ( 2nd ` <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. ) ( ( 1st ` ( Q 2ndF O ) ) ` <. G , P >. ) ) ` ( ( <. F , X >. ( 2nd ` ( Q 2ndF O ) ) <. G , P >. ) ` <. A , K >. ) ) )
106	72, 74	op2nd 7177	. . . . . . . . . . 11 |- ( 2nd ` <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. ) = tpos ( 2nd ` Y )
107	106	oveqi 6663	. . . . . . . . . 10 |- ( ( ( 1st ` ( Q 2ndF O ) ) ` <. F , X >. ) ( 2nd ` <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. ) ( ( 1st ` ( Q 2ndF O ) ) ` <. G , P >. ) ) = ( ( ( 1st ` ( Q 2ndF O ) ) ` <. F , X >. )tpos ( 2nd ` Y ) ( ( 1st ` ( Q 2ndF O ) ) ` <. G , P >. ) )
108		ovtpos 7367	. . . . . . . . . 10 |- ( ( ( 1st ` ( Q 2ndF O ) ) ` <. F , X >. )tpos ( 2nd ` Y ) ( ( 1st ` ( Q 2ndF O ) ) ` <. G , P >. ) ) = ( ( ( 1st ` ( Q 2ndF O ) ) ` <. G , P >. ) ( 2nd ` Y ) ( ( 1st ` ( Q 2ndF O ) ) ` <. F , X >. ) )
109	107, 108	eqtri 2644	. . . . . . . . 9 |- ( ( ( 1st ` ( Q 2ndF O ) ) ` <. F , X >. ) ( 2nd ` <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. ) ( ( 1st ` ( Q 2ndF O ) ) ` <. G , P >. ) ) = ( ( ( 1st ` ( Q 2ndF O ) ) ` <. G , P >. ) ( 2nd ` Y ) ( ( 1st ` ( Q 2ndF O ) ) ` <. F , X >. ) )
110	94, 80	oveq12d 6668	. . . . . . . . 9 |- ( ph -> ( ( ( 1st ` ( Q 2ndF O ) ) ` <. G , P >. ) ( 2nd ` Y ) ( ( 1st ` ( Q 2ndF O ) ) ` <. F , X >. ) ) = ( P ( 2nd ` Y ) X ) )
111	109, 110	syl5eq 2668	. . . . . . . 8 |- ( ph -> ( ( ( 1st ` ( Q 2ndF O ) ) ` <. F , X >. ) ( 2nd ` <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. ) ( ( 1st ` ( Q 2ndF O ) ) ` <. G , P >. ) ) = ( P ( 2nd ` Y ) X ) )
112	7, 13, 55, 26, 18, 27, 50, 54	2ndf2 16836	. . . . . . . . . 10 |- ( ph -> ( <. F , X >. ( 2nd ` ( Q 2ndF O ) ) <. G , P >. ) = ( 2nd |` ( <. F , X >. ( Hom ` ( Q X.c O ) ) <. G , P >. ) ) )
113	112	fveq1d 6193	. . . . . . . . 9 |- ( ph -> ( ( <. F , X >. ( 2nd ` ( Q 2ndF O ) ) <. G , P >. ) ` <. A , K >. ) = ( ( 2nd |` ( <. F , X >. ( Hom ` ( Q X.c O ) ) <. G , P >. ) ) ` <. A , K >. ) )
114		fvres 6207	. . . . . . . . . 10 |- ( <. A , K >. e. ( <. F , X >. ( Hom ` ( Q X.c O ) ) <. G , P >. ) -> ( ( 2nd |` ( <. F , X >. ( Hom ` ( Q X.c O ) ) <. G , P >. ) ) ` <. A , K >. ) = ( 2nd ` <. A , K >. ) )
115	67, 114	syl 17	. . . . . . . . 9 |- ( ph -> ( ( 2nd |` ( <. F , X >. ( Hom ` ( Q X.c O ) ) <. G , P >. ) ) ` <. A , K >. ) = ( 2nd ` <. A , K >. ) )
116		op2ndg 7181	. . . . . . . . . 10 |- ( ( A e. ( F ( O Nat S ) G ) /\ K e. ( P ( Hom ` C ) X ) ) -> ( 2nd ` <. A , K >. ) = K )
117	56, 57, 116	syl2anc 693	. . . . . . . . 9 |- ( ph -> ( 2nd ` <. A , K >. ) = K )
118	113, 115, 117	3eqtrd 2660	. . . . . . . 8 |- ( ph -> ( ( <. F , X >. ( 2nd ` ( Q 2ndF O ) ) <. G , P >. ) ` <. A , K >. ) = K )
119	111, 118	fveq12d 6197	. . . . . . 7 |- ( ph -> ( ( ( ( 1st ` ( Q 2ndF O ) ) ` <. F , X >. ) ( 2nd ` <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. ) ( ( 1st ` ( Q 2ndF O ) ) ` <. G , P >. ) ) ` ( ( <. F , X >. ( 2nd ` ( Q 2ndF O ) ) <. G , P >. ) ` <. A , K >. ) ) = ( ( P ( 2nd ` Y ) X ) ` K ) )
120	105, 119	eqtrd 2656	. . . . . 6 |- ( ph -> ( ( <. F , X >. ( 2nd ` ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ) <. G , P >. ) ` <. A , K >. ) = ( ( P ( 2nd ` Y ) X ) ` K ) )
121	7, 13, 55, 26, 18, 40, 50, 54	1stf2 16833	. . . . . . . 8 |- ( ph -> ( <. F , X >. ( 2nd ` ( Q 1stF O ) ) <. G , P >. ) = ( 1st |` ( <. F , X >. ( Hom ` ( Q X.c O ) ) <. G , P >. ) ) )
122	121	fveq1d 6193	. . . . . . 7 |- ( ph -> ( ( <. F , X >. ( 2nd ` ( Q 1stF O ) ) <. G , P >. ) ` <. A , K >. ) = ( ( 1st |` ( <. F , X >. ( Hom ` ( Q X.c O ) ) <. G , P >. ) ) ` <. A , K >. ) )
123		fvres 6207	. . . . . . . 8 |- ( <. A , K >. e. ( <. F , X >. ( Hom ` ( Q X.c O ) ) <. G , P >. ) -> ( ( 1st |` ( <. F , X >. ( Hom ` ( Q X.c O ) ) <. G , P >. ) ) ` <. A , K >. ) = ( 1st ` <. A , K >. ) )
124	67, 123	syl 17	. . . . . . 7 |- ( ph -> ( ( 1st |` ( <. F , X >. ( Hom ` ( Q X.c O ) ) <. G , P >. ) ) ` <. A , K >. ) = ( 1st ` <. A , K >. ) )
125		op1stg 7180	. . . . . . . 8 |- ( ( A e. ( F ( O Nat S ) G ) /\ K e. ( P ( Hom ` C ) X ) ) -> ( 1st ` <. A , K >. ) = A )
126	56, 57, 125	syl2anc 693	. . . . . . 7 |- ( ph -> ( 1st ` <. A , K >. ) = A )
127	122, 124, 126	3eqtrd 2660	. . . . . 6 |- ( ph -> ( ( <. F , X >. ( 2nd ` ( Q 1stF O ) ) <. G , P >. ) ` <. A , K >. ) = A )
128	120, 127	opeq12d 4410	. . . . 5 |- ( ph -> <. ( ( <. F , X >. ( 2nd ` ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ) <. G , P >. ) ` <. A , K >. ) , ( ( <. F , X >. ( 2nd ` ( Q 1stF O ) ) <. G , P >. ) ` <. A , K >. ) >. = <. ( ( P ( 2nd ` Y ) X ) ` K ) , A >. )
129	104, 128	eqtrd 2656	. . . 4 |- ( ph -> ( ( <. F , X >. ( 2nd ` ( ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ⟨,⟩F ( Q 1stF O ) ) ) <. G , P >. ) ` <. A , K >. ) = <. ( ( P ( 2nd ` Y ) X ) ` K ) , A >. )
130	103, 129	fveq12d 6197	. . 3 |- ( ph -> ( ( ( ( 1st ` ( ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ⟨,⟩F ( Q 1stF O ) ) ) ` <. F , X >. ) ( 2nd ` H ) ( ( 1st ` ( ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ⟨,⟩F ( Q 1stF O ) ) ) ` <. G , P >. ) ) ` ( ( <. F , X >. ( 2nd ` ( ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ⟨,⟩F ( Q 1stF O ) ) ) <. G , P >. ) ` <. A , K >. ) ) = ( ( <. ( ( 1st ` Y ) ` X ) , F >. ( 2nd ` H ) <. ( ( 1st ` Y ) ` P ) , G >. ) ` <. ( ( P ( 2nd ` Y ) X ) ` K ) , A >. ) )
131		df-ov 6653	. . 3 |- ( ( ( P ( 2nd ` Y ) X ) ` K ) ( <. ( ( 1st ` Y ) ` X ) , F >. ( 2nd ` H ) <. ( ( 1st ` Y ) ` P ) , G >. ) A ) = ( ( <. ( ( 1st ` Y ) ` X ) , F >. ( 2nd ` H ) <. ( ( 1st ` Y ) ` P ) , G >. ) ` <. ( ( P ( 2nd ` Y ) X ) ` K ) , A >. )
132	130, 131	syl6eqr 2674	. 2 |- ( ph -> ( ( ( ( 1st ` ( ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ⟨,⟩F ( Q 1stF O ) ) ) ` <. F , X >. ) ( 2nd ` H ) ( ( 1st ` ( ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ⟨,⟩F ( Q 1stF O ) ) ) ` <. G , P >. ) ) ` ( ( <. F , X >. ( 2nd ` ( ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ⟨,⟩F ( Q 1stF O ) ) ) <. G , P >. ) ` <. A , K >. ) ) = ( ( ( P ( 2nd ` Y ) X ) ` K ) ( <. ( ( 1st ` Y ) ` X ) , F >. ( 2nd ` H ) <. ( ( 1st ` Y ) ` P ) , G >. ) A ) )
133	69, 132	eqtrd 2656	1 |- ( ph -> ( A ( <. F , X >. ( 2nd ` Z ) <. G , P >. ) K ) = ( ( ( P ( 2nd ` Y ) X ) ` K ) ( <. ( ( 1st ` Y ) ` X ) , F >. ( 2nd ` H ) <. ( ( 1st ` Y ) ` P ) , G >. ) A ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   C_ wss 3574   <.cop 4183   class class class wbr 4653   X. cxp 5112   ran crn 5115   |` cres 5116   Rel wrel 5119   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167  tpos ctpos 7351   Basecbs 15857   Hom chom 15952   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-cofu 16520  df-nat 16603  df-fuc 16604  df-setc 16726  df-xpc 16812  df-1stf 16813  df-2ndf 16814  df-prf 16815  df-curf 16854  df-hof 16890  df-yon 16891

This theorem is referenced by:  yonedalem3b  16919