Theorem yonedalem4b 16916

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 ) )
yonedalem4b.p	|- ( ph -> P e. B )
yonedalem4b.g	|- ( ph -> G e. ( P ( Hom ` C ) X ) )

Assertion

Ref	Expression
yonedalem4b	|- ( ph -> ( ( ( ( F N X ) ` A ) ` P ) ` G ) = ( ( ( X ( 2nd ` F ) P ) ` G ) ` A ) )

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, G, g, x, y   f, O, g, u, x, y   S, f, g, u, x, y   Q, f, g, u, x   T, f, g, u, y   P, f, g, x, 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)   P( u)   Q( y)   R( x, y, f, g)   T( x)   U( x, y, u, f, g)   .1. ( u)   E( x)   G( u)   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 yonedalem4b

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	20	fveq1d 6193	. . 3 |- ( ph -> ( ( ( F N X ) ` A ) ` P ) = ( ( y e. B |-> ( g e. ( y ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) y ) ` g ) ` A ) ) ) ` P ) )
22	21	fveq1d 6193	. 2 |- ( ph -> ( ( ( ( F N X ) ` A ) ` P ) ` G ) = ( ( ( y e. B |-> ( g e. ( y ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) y ) ` g ) ` A ) ) ) ` P ) ` G ) )
23		eqidd 2623	. . 3 |- ( ph -> ( y e. B |-> ( g e. ( y ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) y ) ` g ) ` A ) ) ) = ( y e. B |-> ( g e. ( y ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) y ) ` g ) ` A ) ) ) )
24		yonedalem4b.p	. . . 4 |- ( ph -> P e. B )
25		ovex 6678	. . . . . 6 |- ( y ( Hom ` C ) X ) e. _V
26	25	mptex 6486	. . . . 5 |- ( g e. ( y ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) y ) ` g ) ` A ) ) e. _V
27	26	a1i 11	. . . 4 |- ( ( ph /\ y = P ) -> ( g e. ( y ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) y ) ` g ) ` A ) ) e. _V )
28		yonedalem4b.g	. . . . . . 7 |- ( ph -> G e. ( P ( Hom ` C ) X ) )
29	28	adantr 481	. . . . . 6 |- ( ( ph /\ y = P ) -> G e. ( P ( Hom ` C ) X ) )
30		simpr 477	. . . . . . 7 |- ( ( ph /\ y = P ) -> y = P )
31	30	oveq1d 6665	. . . . . 6 |- ( ( ph /\ y = P ) -> ( y ( Hom ` C ) X ) = ( P ( Hom ` C ) X ) )
32	29, 31	eleqtrrd 2704	. . . . 5 |- ( ( ph /\ y = P ) -> G e. ( y ( Hom ` C ) X ) )
33		fvexd 6203	. . . . 5 |- ( ( ( ph /\ y = P ) /\ g = G ) -> ( ( ( X ( 2nd ` F ) y ) ` g ) ` A ) e. _V )
34		simplr 792	. . . . . . . 8 |- ( ( ( ph /\ y = P ) /\ g = G ) -> y = P )
35	34	oveq2d 6666	. . . . . . 7 |- ( ( ( ph /\ y = P ) /\ g = G ) -> ( X ( 2nd ` F ) y ) = ( X ( 2nd ` F ) P ) )
36		simpr 477	. . . . . . 7 |- ( ( ( ph /\ y = P ) /\ g = G ) -> g = G )
37	35, 36	fveq12d 6197	. . . . . 6 |- ( ( ( ph /\ y = P ) /\ g = G ) -> ( ( X ( 2nd ` F ) y ) ` g ) = ( ( X ( 2nd ` F ) P ) ` G ) )
38	37	fveq1d 6193	. . . . 5 |- ( ( ( ph /\ y = P ) /\ g = G ) -> ( ( ( X ( 2nd ` F ) y ) ` g ) ` A ) = ( ( ( X ( 2nd ` F ) P ) ` G ) ` A ) )
39	32, 33, 38	fvmptdv2 6298	. . . 4 |- ( ( ph /\ y = P ) -> ( ( ( y e. B |-> ( g e. ( y ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) y ) ` g ) ` A ) ) ) ` P ) = ( g e. ( y ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) y ) ` g ) ` A ) ) -> ( ( ( y e. B |-> ( g e. ( y ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) y ) ` g ) ` A ) ) ) ` P ) ` G ) = ( ( ( X ( 2nd ` F ) P ) ` G ) ` A ) ) )
40		nfmpt1 4747	. . . 4 |- F/_ y ( y e. B |-> ( g e. ( y ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) y ) ` g ) ` A ) ) )
41		nffvmpt1 6199	. . . . . 6 |- F/_ y ( ( y e. B |-> ( g e. ( y ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) y ) ` g ) ` A ) ) ) ` P )
42		nfcv 2764	. . . . . 6 |- F/_ y G
43	41, 42	nffv 6198	. . . . 5 |- F/_ y ( ( ( y e. B |-> ( g e. ( y ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) y ) ` g ) ` A ) ) ) ` P ) ` G )
44	43	nfeq1 2778	. . . 4 |- F/ y ( ( ( y e. B |-> ( g e. ( y ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) y ) ` g ) ` A ) ) ) ` P ) ` G ) = ( ( ( X ( 2nd ` F ) P ) ` G ) ` A )
45	24, 27, 39, 40, 44	fvmptdf 6296	. . 3 |- ( ph -> ( ( y e. B |-> ( g e. ( y ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) y ) ` g ) ` A ) ) ) = ( y e. B |-> ( g e. ( y ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) y ) ` g ) ` A ) ) ) -> ( ( ( y e. B |-> ( g e. ( y ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) y ) ` g ) ` A ) ) ) ` P ) ` G ) = ( ( ( X ( 2nd ` F ) P ) ` G ) ` A ) ) )
46	23, 45	mpd 15	. 2 |- ( ph -> ( ( ( y e. B |-> ( g e. ( y ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) y ) ` g ) ` A ) ) ) ` P ) ` G ) = ( ( ( X ( 2nd ` F ) P ) ` G ) ` A ) )
47	22, 46	eqtrd 2656	1 |- ( ph -> ( ( ( ( F N X ) ` A ) ` P ) ` G ) = ( ( ( X ( 2nd ` F ) P ) ` G ) ` A ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   C_ wss 3574   <.cop 4183   |-> cmpt 4729   ran crn 5115   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   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   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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  df-reu 2919  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-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  yonedalem4c  16917  yonedainv  16921