Theorem yonval 16901

Description: Value of the Yoneda embedding. (Contributed by Mario Carneiro, 17-Jan-2017.)

Hypotheses

Ref	Expression
yonval.y	|- Y = (Yon ` C )
yonval.c	|- ( ph -> C e. Cat )
yonval.o	|- O = (oppCat ` C )
yonval.m	|- M = (HomF ` O )

Assertion

Ref	Expression
yonval	|- ( ph -> Y = ( <. C , O >. curryF M ) )

Proof of Theorem yonval

Dummy variable c is distinct from all other variables.

Step	Hyp	Ref	Expression
1		yonval.y	. 2 |- Y = (Yon ` C )
2		df-yon 16891	. . . 4 |- Yon = ( c e. Cat |-> ( <. c , (oppCat ` c ) >. curryF (HomF ` (oppCat ` c ) ) ) )
3	2	a1i 11	. . 3 |- ( ph -> Yon = ( c e. Cat |-> ( <. c , (oppCat ` c ) >. curryF (HomF ` (oppCat ` c ) ) ) ) )
4		simpr 477	. . . . 5 |- ( ( ph /\ c = C ) -> c = C )
5	4	fveq2d 6195	. . . . . 6 |- ( ( ph /\ c = C ) -> (oppCat ` c ) = (oppCat ` C ) )
6		yonval.o	. . . . . 6 |- O = (oppCat ` C )
7	5, 6	syl6eqr 2674	. . . . 5 |- ( ( ph /\ c = C ) -> (oppCat ` c ) = O )
8	4, 7	opeq12d 4410	. . . 4 |- ( ( ph /\ c = C ) -> <. c , (oppCat ` c ) >. = <. C , O >. )
9	7	fveq2d 6195	. . . . 5 |- ( ( ph /\ c = C ) -> (HomF ` (oppCat ` c ) ) = (HomF ` O ) )
10		yonval.m	. . . . 5 |- M = (HomF ` O )
11	9, 10	syl6eqr 2674	. . . 4 |- ( ( ph /\ c = C ) -> (HomF ` (oppCat ` c ) ) = M )
12	8, 11	oveq12d 6668	. . 3 |- ( ( ph /\ c = C ) -> ( <. c , (oppCat ` c ) >. curryF (HomF ` (oppCat ` c ) ) ) = ( <. C , O >. curryF M ) )
13		yonval.c	. . 3 |- ( ph -> C e. Cat )
14		ovexd 6680	. . 3 |- ( ph -> ( <. C , O >. curryF M ) e. _V )
15	3, 12, 13, 14	fvmptd 6288	. 2 |- ( ph -> (Yon ` C ) = ( <. C , O >. curryF M ) )
16	1, 15	syl5eq 2668	1 |- ( ph -> Y = ( <. C , O >. curryF M ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   Catccat 16325  oppCatcoppc 16371   curry_F ccurf 16850  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-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-ral 2917  df-rex 2918  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-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-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-yon 16891

This theorem is referenced by:  yoncl  16902  yon11  16904  yon12  16905  yon2  16906  yonpropd  16908  oppcyon  16909