Theorem oyoncl 14322

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 ( Homf ` 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 13903	. . . 4 |- ( C e. Cat -> O e. Cat )
5	2, 4	syl 16	. . 3 |- ( ph -> O e. Cat )
6		eqid 2404	. . 3 |- (oppCat ` O ) = (oppCat ` O )
7		oyoncl.s	. . 3 |- S = ( SetCat ` U )
8		eqid 2404	. . 3 |- ( (oppCat ` O ) FuncCat S ) = ( (oppCat ` O ) FuncCat S )
9		oyoncl.u	. . 3 |- ( ph -> U e. V )
10		eqid 2404	. . . . . . 7 |- ( Homf ` C ) = ( Homf ` C )
11	3, 10	oppchomf 13901	. . . . . 6 |- tpos ( Homf ` C ) = ( Homf ` O )
12	11	rneqi 5055	. . . . 5 |- ran tpos ( Homf ` C ) = ran ( Homf ` O )
13		relxp 4942	. . . . . . 7 |- Rel ( ( Base ` C ) X. ( Base ` C ) )
14		eqid 2404	. . . . . . . . . 10 |- ( Base ` C ) = ( Base ` C )
15	10, 14	homffn 13874	. . . . . . . . 9 |- ( Homf ` C ) Fn ( ( Base ` C ) X. ( Base ` C ) )
16		fndm 5503	. . . . . . . . 9 |- ( ( Homf ` C ) Fn ( ( Base ` C ) X. ( Base ` C ) ) -> dom ( Homf ` C ) = ( ( Base ` C ) X. ( Base ` C ) ) )
17	15, 16	ax-mp 8	. . . . . . . 8 |- dom ( Homf ` C ) = ( ( Base ` C ) X. ( Base ` C ) )
18	17	releqi 4919	. . . . . . 7 |- ( Rel dom ( Homf ` C ) <-> Rel ( ( Base ` C ) X. ( Base ` C ) ) )
19	13, 18	mpbir 201	. . . . . 6 |- Rel dom ( Homf ` C )
20		rntpos 6451	. . . . . 6 |- ( Rel dom ( Homf ` C ) -> ran tpos ( Homf ` C ) = ran ( Homf ` C ) )
21	19, 20	ax-mp 8	. . . . 5 |- ran tpos ( Homf ` C ) = ran ( Homf ` C )
22	12, 21	eqtr3i 2426	. . . 4 |- ran ( Homf ` O ) = ran ( Homf ` C )
23		oyoncl.h	. . . 4 |- ( ph -> ran ( Homf ` C ) C_ U )
24	22, 23	syl5eqss 3352	. . 3 |- ( ph -> ran ( Homf ` O ) C_ U )
25	1, 5, 6, 7, 8, 9, 24	yoncl 14314	. 2 |- ( ph -> Y e. ( O Func ( (oppCat ` O ) FuncCat S ) ) )
26		oyoncl.q	. . . 4 |- Q = ( C FuncCat S )
27	3	2oppchomf 13905	. . . . . 6 |- ( Homf ` C ) = ( Homf ` (oppCat ` O ) )
28	27	a1i 11	. . . . 5 |- ( ph -> ( Homf ` C ) = ( Homf ` (oppCat ` O ) ) )
29	3	2oppccomf 13906	. . . . . 6 |- (compf ` C ) = (compf ` (oppCat ` O ) )
30	29	a1i 11	. . . . 5 |- ( ph -> (compf ` C ) = (compf ` (oppCat ` O ) ) )
31		eqidd 2405	. . . . 5 |- ( ph -> ( Homf ` S ) = ( Homf ` S ) )
32		eqidd 2405	. . . . 5 |- ( ph -> (compf ` S ) = (compf ` S ) )
33	6	oppccat 13903	. . . . . 6 |- ( O e. Cat -> (oppCat ` O ) e. Cat )
34	5, 33	syl 16	. . . . 5 |- ( ph -> (oppCat ` O ) e. Cat )
35	7	setccat 14195	. . . . . 6 |- ( U e. V -> S e. Cat )
36	9, 35	syl 16	. . . . 5 |- ( ph -> S e. Cat )
37	28, 30, 31, 32, 2, 34, 36, 36	fucpropd 14129	. . . 4 |- ( ph -> ( C FuncCat S ) = ( (oppCat ` O ) FuncCat S ) )
38	26, 37	syl5eq 2448	. . 3 |- ( ph -> Q = ( (oppCat ` O ) FuncCat S ) )
39	38	oveq2d 6056	. 2 |- ( ph -> ( O Func Q ) = ( O Func ( (oppCat ` O ) FuncCat S ) ) )
40	25, 39	eleqtrrd 2481	1 |- ( ph -> Y e. ( O Func Q ) )

Syntax hints:   -> wi 4   = wceq 1649   e. wcel 1721   C_ wss 3280   X. cxp 4835   dom cdm 4837   ran crn 4838   Rel wrel 4842   Fn wfn 5408   ` cfv 5413  (class class class)co 6040  tpos ctpos 6437   Basecbs 13424   Catccat 13844   Hom_f chomf 13846  comp_fccomf 13847  oppCatcoppc 13892   Func cfunc 14006   FuncCat cfuc 14094   SetCatcsetc 14185  Yoncyon 14301

This theorem is referenced by:  oyon1cl  14323

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-tpos 6438  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-fz 11000  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-hom 13508  df-cco 13509  df-cat 13848  df-cid 13849  df-homf 13850  df-comf 13851  df-oppc 13893  df-func 14010  df-nat 14095  df-fuc 14096  df-setc 14186  df-xpc 14224  df-curf 14266  df-hof 14302  df-yon 14303