Theorem oyoncl 15076

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 14657	. . . 4 |- ( C e. Cat -> O e. Cat )
5	2, 4	syl 16	. . 3 |- ( ph -> O e. Cat )
6		eqid 2441	. . 3 |- (oppCat ` O ) = (oppCat ` O )
7		oyoncl.s	. . 3 |- S = ( SetCat ` U )
8		eqid 2441	. . 3 |- ( (oppCat ` O ) FuncCat S ) = ( (oppCat ` O ) FuncCat S )
9		oyoncl.u	. . 3 |- ( ph -> U e. V )
10		eqid 2441	. . . . . . 7 |- ( Hom f ` C ) = ( Hom f ` C )
11	3, 10	oppchomf 14655	. . . . . 6 |- tpos ( Hom f ` C ) = ( Hom f ` O )
12	11	rneqi 5062	. . . . 5 |- ran tpos ( Hom f ` C ) = ran ( Hom f ` O )
13		relxp 4943	. . . . . . 7 |- Rel ( ( Base ` C ) X. ( Base ` C ) )
14		eqid 2441	. . . . . . . . . 10 |- ( Base ` C ) = ( Base ` C )
15	10, 14	homffn 14628	. . . . . . . . 9 |- ( Hom f ` C ) Fn ( ( Base ` C ) X. ( Base ` C ) )
16		fndm 5507	. . . . . . . . 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 4919	. . . . . . 7 |- ( Rel dom ( Hom f ` C ) <-> Rel ( ( Base ` C ) X. ( Base ` C ) ) )
19	13, 18	mpbir 209	. . . . . 6 |- Rel dom ( Hom f ` C )
20		rntpos 6757	. . . . . 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 2463	. . . 4 |- ran ( Hom f ` O ) = ran ( Hom f ` C )
23		oyoncl.h	. . . 4 |- ( ph -> ran ( Hom f ` C ) C_ U )
24	22, 23	syl5eqss 3397	. . 3 |- ( ph -> ran ( Hom f ` O ) C_ U )
25	1, 5, 6, 7, 8, 9, 24	yoncl 15068	. 2 |- ( ph -> Y e. ( O Func ( (oppCat ` O ) FuncCat S ) ) )
26		oyoncl.q	. . . 4 |- Q = ( C FuncCat S )
27	3	2oppchomf 14659	. . . . . 6 |- ( Hom f ` C ) = ( Hom f ` (oppCat ` O ) )
28	27	a1i 11	. . . . 5 |- ( ph -> ( Hom f ` C ) = ( Hom f ` (oppCat ` O ) ) )
29	3	2oppccomf 14660	. . . . . 6 |- (compf ` C ) = (compf ` (oppCat ` O ) )
30	29	a1i 11	. . . . 5 |- ( ph -> (compf ` C ) = (compf ` (oppCat ` O ) ) )
31		eqidd 2442	. . . . 5 |- ( ph -> ( Hom f ` S ) = ( Hom f ` S ) )
32		eqidd 2442	. . . . 5 |- ( ph -> (compf ` S ) = (compf ` S ) )
33	6	oppccat 14657	. . . . . 6 |- ( O e. Cat -> (oppCat ` O ) e. Cat )
34	5, 33	syl 16	. . . . 5 |- ( ph -> (oppCat ` O ) e. Cat )
35	7	setccat 14949	. . . . . 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 14883	. . . 4 |- ( ph -> ( C FuncCat S ) = ( (oppCat ` O ) FuncCat S ) )
38	26, 37	syl5eq 2485	. . 3 |- ( ph -> Q = ( (oppCat ` O ) FuncCat S ) )
39	38	oveq2d 6106	. 2 |- ( ph -> ( O Func Q ) = ( O Func ( (oppCat ` O ) FuncCat S ) ) )
40	25, 39	eleqtrrd 2518	1 |- ( ph -> Y e. ( O Func Q ) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 1761   C_ wss 3325   X. cxp 4834   dom cdm 4836   ran crn 4837   Rel wrel 4841   Fn wfn 5410   ` cfv 5415  (class class class)co 6090  tpos ctpos 6743   Basecbs 14170   Catccat 14598   Hom _f chomf 14600  comp_fccomf 14601  oppCatcoppc 14646   Func cfunc 14760   FuncCat cfuc 14848   SetCatcsetc 14939  Yoncyon 15055

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-tpos 6744  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-fz 11434  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-hom 14258  df-cco 14259  df-cat 14602  df-cid 14603  df-homf 14604  df-comf 14605  df-oppc 14647  df-func 14764  df-nat 14849  df-fuc 14850  df-setc 14940  df-xpc 14978  df-curf 15020  df-hof 15056  df-yon 15057

This theorem is referenced by:  oyon1cl  15077