Theorem yon12 16905

Description: Value of the Yoneda embedding at a morphism. The partially evaluated Yoneda embedding is also the contravariant Hom functor. (Contributed by Mario Carneiro, 17-Jan-2017.)

Hypotheses

Ref	Expression
yon11.y	|- Y = (Yon ` C )
yon11.b	|- B = ( Base ` C )
yon11.c	|- ( ph -> C e. Cat )
yon11.p	|- ( ph -> X e. B )
yon11.h	|- H = ( Hom ` C )
yon11.z	|- ( ph -> Z e. B )
yon12.x	|- .x. = (comp ` C )
yon12.w	|- ( ph -> W e. B )
yon12.f	|- ( ph -> F e. ( W H Z ) )
yon12.g	|- ( ph -> G e. ( Z H X ) )

Assertion

Ref	Expression
yon12	|- ( ph -> ( ( ( Z ( 2nd ` ( ( 1st ` Y ) ` X ) ) W ) ` F ) ` G ) = ( G ( <. W , Z >. .x. X ) F ) )

Proof of Theorem yon12

Step	Hyp	Ref	Expression
1		yon11.y	. . . . . . . . . 10 |- Y = (Yon ` C )
2		yon11.c	. . . . . . . . . 10 |- ( ph -> C e. Cat )
3		eqid 2622	. . . . . . . . . 10 |- (oppCat ` C ) = (oppCat ` C )
4		eqid 2622	. . . . . . . . . 10 |- (HomF ` (oppCat ` C ) ) = (HomF ` (oppCat ` C ) )
5	1, 2, 3, 4	yonval 16901	. . . . . . . . 9 |- ( ph -> Y = ( <. C , (oppCat ` C ) >. curryF (HomF ` (oppCat ` C ) ) ) )
6	5	fveq2d 6195	. . . . . . . 8 |- ( ph -> ( 1st ` Y ) = ( 1st ` ( <. C , (oppCat ` C ) >. curryF (HomF ` (oppCat ` C ) ) ) ) )
7	6	fveq1d 6193	. . . . . . 7 |- ( ph -> ( ( 1st ` Y ) ` X ) = ( ( 1st ` ( <. C , (oppCat ` C ) >. curryF (HomF ` (oppCat ` C ) ) ) ) ` X ) )
8	7	fveq2d 6195	. . . . . 6 |- ( ph -> ( 2nd ` ( ( 1st ` Y ) ` X ) ) = ( 2nd ` ( ( 1st ` ( <. C , (oppCat ` C ) >. curryF (HomF ` (oppCat ` C ) ) ) ) ` X ) ) )
9	8	oveqd 6667	. . . . 5 |- ( ph -> ( Z ( 2nd ` ( ( 1st ` Y ) ` X ) ) W ) = ( Z ( 2nd ` ( ( 1st ` ( <. C , (oppCat ` C ) >. curryF (HomF ` (oppCat ` C ) ) ) ) ` X ) ) W ) )
10	9	fveq1d 6193	. . . 4 |- ( ph -> ( ( Z ( 2nd ` ( ( 1st ` Y ) ` X ) ) W ) ` F ) = ( ( Z ( 2nd ` ( ( 1st ` ( <. C , (oppCat ` C ) >. curryF (HomF ` (oppCat ` C ) ) ) ) ` X ) ) W ) ` F ) )
11		eqid 2622	. . . . 5 |- ( <. C , (oppCat ` C ) >. curryF (HomF ` (oppCat ` C ) ) ) = ( <. C , (oppCat ` C ) >. curryF (HomF ` (oppCat ` C ) ) )
12		yon11.b	. . . . 5 |- B = ( Base ` C )
13	3	oppccat 16382	. . . . . 6 |- ( C e. Cat -> (oppCat ` C ) e. Cat )
14	2, 13	syl 17	. . . . 5 |- ( ph -> (oppCat ` C ) e. Cat )
15		eqid 2622	. . . . . 6 |- ( SetCat ` ran ( Hom f ` C ) ) = ( SetCat ` ran ( Hom f ` C ) )
16		fvex 6201	. . . . . . . 8 |- ( Hom f ` C ) e. _V
17	16	rnex 7100	. . . . . . 7 |- ran ( Hom f ` C ) e. _V
18	17	a1i 11	. . . . . 6 |- ( ph -> ran ( Hom f ` C ) e. _V )
19		ssid 3624	. . . . . . 7 |- ran ( Hom f ` C ) C_ ran ( Hom f ` C )
20	19	a1i 11	. . . . . 6 |- ( ph -> ran ( Hom f ` C ) C_ ran ( Hom f ` C ) )
21	3, 4, 15, 2, 18, 20	oppchofcl 16900	. . . . 5 |- ( ph -> (HomF ` (oppCat ` C ) ) e. ( ( C X.c (oppCat ` C ) ) Func ( SetCat ` ran ( Hom f ` C ) ) ) )
22	3, 12	oppcbas 16378	. . . . 5 |- B = ( Base ` (oppCat ` C ) )
23		yon11.p	. . . . 5 |- ( ph -> X e. B )
24		eqid 2622	. . . . 5 |- ( ( 1st ` ( <. C , (oppCat ` C ) >. curryF (HomF ` (oppCat ` C ) ) ) ) ` X ) = ( ( 1st ` ( <. C , (oppCat ` C ) >. curryF (HomF ` (oppCat ` C ) ) ) ) ` X )
25		yon11.z	. . . . 5 |- ( ph -> Z e. B )
26		eqid 2622	. . . . 5 |- ( Hom ` (oppCat ` C ) ) = ( Hom ` (oppCat ` C ) )
27		eqid 2622	. . . . 5 |- ( Id ` C ) = ( Id ` C )
28		yon12.w	. . . . 5 |- ( ph -> W e. B )
29		yon12.f	. . . . . 6 |- ( ph -> F e. ( W H Z ) )
30		yon11.h	. . . . . . 7 |- H = ( Hom ` C )
31	30, 3	oppchom 16375	. . . . . 6 |- ( Z ( Hom ` (oppCat ` C ) ) W ) = ( W H Z )
32	29, 31	syl6eleqr 2712	. . . . 5 |- ( ph -> F e. ( Z ( Hom ` (oppCat ` C ) ) W ) )
33	11, 12, 2, 14, 21, 22, 23, 24, 25, 26, 27, 28, 32	curf12 16867	. . . 4 |- ( ph -> ( ( Z ( 2nd ` ( ( 1st ` ( <. C , (oppCat ` C ) >. curryF (HomF ` (oppCat ` C ) ) ) ) ` X ) ) W ) ` F ) = ( ( ( Id ` C ) ` X ) ( <. X , Z >. ( 2nd ` (HomF ` (oppCat ` C ) ) ) <. X , W >. ) F ) )
34	10, 33	eqtrd 2656	. . 3 |- ( ph -> ( ( Z ( 2nd ` ( ( 1st ` Y ) ` X ) ) W ) ` F ) = ( ( ( Id ` C ) ` X ) ( <. X , Z >. ( 2nd ` (HomF ` (oppCat ` C ) ) ) <. X , W >. ) F ) )
35	34	fveq1d 6193	. 2 |- ( ph -> ( ( ( Z ( 2nd ` ( ( 1st ` Y ) ` X ) ) W ) ` F ) ` G ) = ( ( ( ( Id ` C ) ` X ) ( <. X , Z >. ( 2nd ` (HomF ` (oppCat ` C ) ) ) <. X , W >. ) F ) ` G ) )
36		eqid 2622	. . 3 |- (comp ` (oppCat ` C ) ) = (comp ` (oppCat ` C ) )
37	12, 30, 27, 2, 23	catidcl 16343	. . . 4 |- ( ph -> ( ( Id ` C ) ` X ) e. ( X H X ) )
38	30, 3	oppchom 16375	. . . 4 |- ( X ( Hom ` (oppCat ` C ) ) X ) = ( X H X )
39	37, 38	syl6eleqr 2712	. . 3 |- ( ph -> ( ( Id ` C ) ` X ) e. ( X ( Hom ` (oppCat ` C ) ) X ) )
40		yon12.g	. . . 4 |- ( ph -> G e. ( Z H X ) )
41	30, 3	oppchom 16375	. . . 4 |- ( X ( Hom ` (oppCat ` C ) ) Z ) = ( Z H X )
42	40, 41	syl6eleqr 2712	. . 3 |- ( ph -> G e. ( X ( Hom ` (oppCat ` C ) ) Z ) )
43	4, 14, 22, 26, 23, 25, 23, 28, 36, 39, 32, 42	hof2 16897	. 2 |- ( ph -> ( ( ( ( Id ` C ) ` X ) ( <. X , Z >. ( 2nd ` (HomF ` (oppCat ` C ) ) ) <. X , W >. ) F ) ` G ) = ( ( F ( <. X , Z >. (comp ` (oppCat ` C ) ) W ) G ) ( <. X , X >. (comp ` (oppCat ` C ) ) W ) ( ( Id ` C ) ` X ) ) )
44		yon12.x	. . . . 5 |- .x. = (comp ` C )
45	12, 44, 3, 23, 25, 28	oppcco 16377	. . . 4 |- ( ph -> ( F ( <. X , Z >. (comp ` (oppCat ` C ) ) W ) G ) = ( G ( <. W , Z >. .x. X ) F ) )
46	45	oveq1d 6665	. . 3 |- ( ph -> ( ( F ( <. X , Z >. (comp ` (oppCat ` C ) ) W ) G ) ( <. X , X >. (comp ` (oppCat ` C ) ) W ) ( ( Id ` C ) ` X ) ) = ( ( G ( <. W , Z >. .x. X ) F ) ( <. X , X >. (comp ` (oppCat ` C ) ) W ) ( ( Id ` C ) ` X ) ) )
47	12, 44, 3, 23, 23, 28	oppcco 16377	. . 3 |- ( ph -> ( ( G ( <. W , Z >. .x. X ) F ) ( <. X , X >. (comp ` (oppCat ` C ) ) W ) ( ( Id ` C ) ` X ) ) = ( ( ( Id ` C ) ` X ) ( <. W , X >. .x. X ) ( G ( <. W , Z >. .x. X ) F ) ) )
48	12, 30, 44, 2, 28, 25, 23, 29, 40	catcocl 16346	. . . 4 |- ( ph -> ( G ( <. W , Z >. .x. X ) F ) e. ( W H X ) )
49	12, 30, 27, 2, 28, 44, 23, 48	catlid 16344	. . 3 |- ( ph -> ( ( ( Id ` C ) ` X ) ( <. W , X >. .x. X ) ( G ( <. W , Z >. .x. X ) F ) ) = ( G ( <. W , Z >. .x. X ) F ) )
50	46, 47, 49	3eqtrd 2660	. 2 |- ( ph -> ( ( F ( <. X , Z >. (comp ` (oppCat ` C ) ) W ) G ) ( <. X , X >. (comp ` (oppCat ` C ) ) W ) ( ( Id ` C ) ` X ) ) = ( G ( <. W , Z >. .x. X ) F ) )
51	35, 43, 50	3eqtrd 2660	1 |- ( ph -> ( ( ( Z ( 2nd ` ( ( 1st ` Y ) ` X ) ) W ) ` F ) ` G ) = ( G ( <. W , Z >. .x. X ) F ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   <.cop 4183   ran crn 5115   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   Basecbs 15857   Hom chom 15952  compcco 15953   Catccat 16325   Idccid 16326   Hom _f chomf 16327  oppCatcoppc 16371   SetCatcsetc 16725   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-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-setc 16726  df-xpc 16812  df-curf 16854  df-hof 16890  df-yon 16891

This theorem is referenced by:  yonedalem4c  16917  yonedalem3b  16919  yonedainv  16921  yonffthlem  16922