Theorem oyoncl 16910

Description: The opposite Yoneda embedding is a functor from oppCat ` C to the functor category C -> SetCat. (Contributed by Mario Carneiro, 26-Jan-2017.)

Hypotheses

Ref	Expression
oyoncl.o	|- O = (oppCat ` C )
oyoncl.y	|- Y = (Yon ` O )
oyoncl.c	|- ( ph -> C e. Cat )
oyoncl.s	|- S = ( SetCat ` U )
oyoncl.u	|- ( ph -> U e. V )
oyoncl.h	|- ( ph -> ran ( Hom f ` C ) C_ U )
oyoncl.q	|- Q = ( C FuncCat S )

Assertion

Ref	Expression
oyoncl	|- ( ph -> Y e. ( O Func Q ) )

Proof of Theorem oyoncl

Step	Hyp	Ref	Expression
1		oyoncl.y	. . 3 |- Y = (Yon ` O )
2		oyoncl.c	. . . 4 |- ( ph -> C e. Cat )
3		oyoncl.o	. . . . 5 |- O = (oppCat ` C )
4	3	oppccat 16382	. . . 4 |- ( C e. Cat -> O e. Cat )
5	2, 4	syl 17	. . 3 |- ( ph -> O e. Cat )
6		eqid 2622	. . 3 |- (oppCat ` O ) = (oppCat ` O )
7		oyoncl.s	. . 3 |- S = ( SetCat ` U )
8		eqid 2622	. . 3 |- ( (oppCat ` O ) FuncCat S ) = ( (oppCat ` O ) FuncCat S )
9		oyoncl.u	. . 3 |- ( ph -> U e. V )
10		eqid 2622	. . . . . . 7 |- ( Hom f ` C ) = ( Hom f ` C )
11	3, 10	oppchomf 16380	. . . . . 6 |- tpos ( Hom f ` C ) = ( Hom f ` O )
12	11	rneqi 5352	. . . . 5 |- ran tpos ( Hom f ` C ) = ran ( Hom f ` O )
13		relxp 5227	. . . . . . 7 |- Rel ( ( Base ` C ) X. ( Base ` C ) )
14		eqid 2622	. . . . . . . . . 10 |- ( Base ` C ) = ( Base ` C )
15	10, 14	homffn 16353	. . . . . . . . 9 |- ( Hom f ` C ) Fn ( ( Base ` C ) X. ( Base ` C ) )
16		fndm 5990	. . . . . . . . 9 |- ( ( Hom f ` C ) Fn ( ( Base ` C ) X. ( Base ` C ) ) -> dom ( Hom f ` C ) = ( ( Base ` C ) X. ( Base ` C ) ) )
17	15, 16	ax-mp 5	. . . . . . . 8 |- dom ( Hom f ` C ) = ( ( Base ` C ) X. ( Base ` C ) )
18	17	releqi 5202	. . . . . . 7 |- ( Rel dom ( Hom f ` C ) <-> Rel ( ( Base ` C ) X. ( Base ` C ) ) )
19	13, 18	mpbir 221	. . . . . 6 |- Rel dom ( Hom f ` C )
20		rntpos 7365	. . . . . 6 |- ( Rel dom ( Hom f ` C ) -> ran tpos ( Hom f ` C ) = ran ( Hom f ` C ) )
21	19, 20	ax-mp 5	. . . . 5 |- ran tpos ( Hom f ` C ) = ran ( Hom f ` C )
22	12, 21	eqtr3i 2646	. . . 4 |- ran ( Hom f ` O ) = ran ( Hom f ` C )
23		oyoncl.h	. . . 4 |- ( ph -> ran ( Hom f ` C ) C_ U )
24	22, 23	syl5eqss 3649	. . 3 |- ( ph -> ran ( Hom f ` O ) C_ U )
25	1, 5, 6, 7, 8, 9, 24	yoncl 16902	. 2 |- ( ph -> Y e. ( O Func ( (oppCat ` O ) FuncCat S ) ) )
26		oyoncl.q	. . . 4 |- Q = ( C FuncCat S )
27	3	2oppchomf 16384	. . . . . 6 |- ( Hom f ` C ) = ( Hom f ` (oppCat ` O ) )
28	27	a1i 11	. . . . 5 |- ( ph -> ( Hom f ` C ) = ( Hom f ` (oppCat ` O ) ) )
29	3	2oppccomf 16385	. . . . . 6 |- (compf ` C ) = (compf ` (oppCat ` O ) )
30	29	a1i 11	. . . . 5 |- ( ph -> (compf ` C ) = (compf ` (oppCat ` O ) ) )
31		eqidd 2623	. . . . 5 |- ( ph -> ( Hom f ` S ) = ( Hom f ` S ) )
32		eqidd 2623	. . . . 5 |- ( ph -> (compf ` S ) = (compf ` S ) )
33	6	oppccat 16382	. . . . . 6 |- ( O e. Cat -> (oppCat ` O ) e. Cat )
34	5, 33	syl 17	. . . . 5 |- ( ph -> (oppCat ` O ) e. Cat )
35	7	setccat 16735	. . . . . 6 |- ( U e. V -> S e. Cat )
36	9, 35	syl 17	. . . . 5 |- ( ph -> S e. Cat )
37	28, 30, 31, 32, 2, 34, 36, 36	fucpropd 16637	. . . 4 |- ( ph -> ( C FuncCat S ) = ( (oppCat ` O ) FuncCat S ) )
38	26, 37	syl5eq 2668	. . 3 |- ( ph -> Q = ( (oppCat ` O ) FuncCat S ) )
39	38	oveq2d 6666	. 2 |- ( ph -> ( O Func Q ) = ( O Func ( (oppCat ` O ) FuncCat S ) ) )
40	25, 39	eleqtrrd 2704	1 |- ( ph -> Y e. ( O Func Q ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   C_ wss 3574   X. cxp 5112   dom cdm 5114   ran crn 5115   Rel wrel 5119   Fn wfn 5883   ` cfv 5888  (class class class)co 6650  tpos ctpos 7351   Basecbs 15857   Catccat 16325   Hom _f chomf 16327  comp_fccomf 16328  oppCatcoppc 16371   Func cfunc 16514   FuncCat cfuc 16602   SetCatcsetc 16725  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-nat 16603  df-fuc 16604  df-setc 16726  df-xpc 16812  df-curf 16854  df-hof 16890  df-yon 16891

This theorem is referenced by:  oyon1cl  16911