Theorem yonedalem4a 16915

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
yonedalem4a	|- ( ph -> ( ( F N X ) ` A ) = ( y e. B |-> ( g e. ( y ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) y ) ` 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, 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 yonedalem4a

Step	Hyp	Ref	Expression
1		yonedalem4.n	. . . 4 |- 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 ) ) ) ) )
2	1	a1i 11	. . 3 |- ( ph -> 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 ) ) ) ) ) )
3		simprl 794	. . . . . 6 |- ( ( ph /\ ( f = F /\ x = X ) ) -> f = F )
4	3	fveq2d 6195	. . . . 5 |- ( ( ph /\ ( f = F /\ x = X ) ) -> ( 1st ` f ) = ( 1st ` F ) )
5		simprr 796	. . . . 5 |- ( ( ph /\ ( f = F /\ x = X ) ) -> x = X )
6	4, 5	fveq12d 6197	. . . 4 |- ( ( ph /\ ( f = F /\ x = X ) ) -> ( ( 1st ` f ) ` x ) = ( ( 1st ` F ) ` X ) )
7		simplrr 801	. . . . . . 7 |- ( ( ( ph /\ ( f = F /\ x = X ) ) /\ y e. B ) -> x = X )
8	7	oveq2d 6666	. . . . . 6 |- ( ( ( ph /\ ( f = F /\ x = X ) ) /\ y e. B ) -> ( y ( Hom ` C ) x ) = ( y ( Hom ` C ) X ) )
9		simplrl 800	. . . . . . . . . 10 |- ( ( ( ph /\ ( f = F /\ x = X ) ) /\ y e. B ) -> f = F )
10	9	fveq2d 6195	. . . . . . . . 9 |- ( ( ( ph /\ ( f = F /\ x = X ) ) /\ y e. B ) -> ( 2nd ` f ) = ( 2nd ` F ) )
11		eqidd 2623	. . . . . . . . 9 |- ( ( ( ph /\ ( f = F /\ x = X ) ) /\ y e. B ) -> y = y )
12	10, 7, 11	oveq123d 6671	. . . . . . . 8 |- ( ( ( ph /\ ( f = F /\ x = X ) ) /\ y e. B ) -> ( x ( 2nd ` f ) y ) = ( X ( 2nd ` F ) y ) )
13	12	fveq1d 6193	. . . . . . 7 |- ( ( ( ph /\ ( f = F /\ x = X ) ) /\ y e. B ) -> ( ( x ( 2nd ` f ) y ) ` g ) = ( ( X ( 2nd ` F ) y ) ` g ) )
14	13	fveq1d 6193	. . . . . 6 |- ( ( ( ph /\ ( f = F /\ x = X ) ) /\ y e. B ) -> ( ( ( x ( 2nd ` f ) y ) ` g ) ` u ) = ( ( ( X ( 2nd ` F ) y ) ` g ) ` u ) )
15	8, 14	mpteq12dv 4733	. . . . 5 |- ( ( ( ph /\ ( f = F /\ x = X ) ) /\ y e. B ) -> ( g e. ( y ( Hom ` C ) x ) |-> ( ( ( x ( 2nd ` f ) y ) ` g ) ` u ) ) = ( g e. ( y ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) y ) ` g ) ` u ) ) )
16	15	mpteq2dva 4744	. . . 4 |- ( ( ph /\ ( f = F /\ x = X ) ) -> ( y e. B |-> ( g e. ( y ( Hom ` C ) x ) |-> ( ( ( x ( 2nd ` f ) y ) ` g ) ` u ) ) ) = ( y e. B |-> ( g e. ( y ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) y ) ` g ) ` u ) ) ) )
17	6, 16	mpteq12dv 4733	. . 3 |- ( ( ph /\ ( f = F /\ x = X ) ) -> ( u e. ( ( 1st ` f ) ` x ) |-> ( y e. B |-> ( g e. ( y ( Hom ` C ) x ) |-> ( ( ( x ( 2nd ` f ) y ) ` g ) ` u ) ) ) ) = ( u e. ( ( 1st ` F ) ` X ) |-> ( y e. B |-> ( g e. ( y ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) y ) ` g ) ` u ) ) ) ) )
18		yonedalem21.f	. . 3 |- ( ph -> F e. ( O Func S ) )
19		yonedalem21.x	. . 3 |- ( ph -> X e. B )
20		fvex 6201	. . . . 5 |- ( ( 1st ` F ) ` X ) e. _V
21	20	mptex 6486	. . . 4 |- ( u e. ( ( 1st ` F ) ` X ) |-> ( y e. B |-> ( g e. ( y ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) y ) ` g ) ` u ) ) ) ) e. _V
22	21	a1i 11	. . 3 |- ( ph -> ( u e. ( ( 1st ` F ) ` X ) |-> ( y e. B |-> ( g e. ( y ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) y ) ` g ) ` u ) ) ) ) e. _V )
23	2, 17, 18, 19, 22	ovmpt2d 6788	. 2 |- ( ph -> ( F N X ) = ( u e. ( ( 1st ` F ) ` X ) |-> ( y e. B |-> ( g e. ( y ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) y ) ` g ) ` u ) ) ) ) )
24		simpr 477	. . . . 5 |- ( ( ph /\ u = A ) -> u = A )
25	24	fveq2d 6195	. . . 4 |- ( ( ph /\ u = A ) -> ( ( ( X ( 2nd ` F ) y ) ` g ) ` u ) = ( ( ( X ( 2nd ` F ) y ) ` g ) ` A ) )
26	25	mpteq2dv 4745	. . 3 |- ( ( ph /\ u = A ) -> ( g e. ( y ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) y ) ` g ) ` u ) ) = ( g e. ( y ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) y ) ` g ) ` A ) ) )
27	26	mpteq2dv 4745	. 2 |- ( ( ph /\ u = A ) -> ( y e. B |-> ( g e. ( y ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) y ) ` g ) ` u ) ) ) = ( y e. B |-> ( g e. ( y ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) y ) ` g ) ` A ) ) ) )
28		yonedalem4.p	. 2 |- ( ph -> A e. ( ( 1st ` F ) ` X ) )
29		yoneda.b	. . . . 5 |- B = ( Base ` C )
30		fvex 6201	. . . . 5 |- ( Base ` C ) e. _V
31	29, 30	eqeltri 2697	. . . 4 |- B e. _V
32	31	mptex 6486	. . 3 |- ( y e. B |-> ( g e. ( y ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) y ) ` g ) ` A ) ) ) e. _V
33	32	a1i 11	. 2 |- ( ph -> ( y e. B |-> ( g e. ( y ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) y ) ` g ) ` A ) ) ) e. _V )
34	23, 27, 28, 33	fvmptd 6288	1 |- ( ph -> ( ( F N X ) ` A ) = ( y e. B |-> ( g e. ( y ( Hom ` C ) X ) |-> ( ( ( X ( 2nd ` F ) y ) ` 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-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-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:  yonedalem4b  16916  yonedalem4c  16917  yonffthlem  16922