Theorem oyoncl 15414

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 14995	. . . 4 |- ( C e. Cat -> O e. Cat )
5	2, 4	syl 16	. . 3 |- ( ph -> O e. Cat )
6		eqid 2467	. . 3 |- (oppCat ` O ) = (oppCat ` O )
7		oyoncl.s	. . 3 |- S = ( SetCat ` U )
8		eqid 2467	. . 3 |- ( (oppCat ` O ) FuncCat S ) = ( (oppCat ` O ) FuncCat S )
9		oyoncl.u	. . 3 |- ( ph -> U e. V )
10		eqid 2467	. . . . . . 7 |- ( Hom f ` C ) = ( Hom f ` C )
11	3, 10	oppchomf 14993	. . . . . 6 |- tpos ( Hom f ` C ) = ( Hom f ` O )
12	11	rneqi 5235	. . . . 5 |- ran tpos ( Hom f ` C ) = ran ( Hom f ` O )
13		relxp 5116	. . . . . . 7 |- Rel ( ( Base ` C ) X. ( Base ` C ) )
14		eqid 2467	. . . . . . . . . 10 |- ( Base ` C ) = ( Base ` C )
15	10, 14	homffn 14966	. . . . . . . . 9 |- ( Hom f ` C ) Fn ( ( Base ` C ) X. ( Base ` C ) )
16		fndm 5686	. . . . . . . . 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 5092	. . . . . . 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 6980	. . . . . 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 2498	. . . 4 |- ran ( Hom f ` O ) = ran ( Hom f ` C )
23		oyoncl.h	. . . 4 |- ( ph -> ran ( Hom f ` C ) C_ U )
24	22, 23	syl5eqss 3553	. . 3 |- ( ph -> ran ( Hom f ` O ) C_ U )
25	1, 5, 6, 7, 8, 9, 24	yoncl 15406	. 2 |- ( ph -> Y e. ( O Func ( (oppCat ` O ) FuncCat S ) ) )
26		oyoncl.q	. . . 4 |- Q = ( C FuncCat S )
27	3	2oppchomf 14997	. . . . . 6 |- ( Hom f ` C ) = ( Hom f ` (oppCat ` O ) )
28	27	a1i 11	. . . . 5 |- ( ph -> ( Hom f ` C ) = ( Hom f ` (oppCat ` O ) ) )
29	3	2oppccomf 14998	. . . . . 6 |- (compf ` C ) = (compf ` (oppCat ` O ) )
30	29	a1i 11	. . . . 5 |- ( ph -> (compf ` C ) = (compf ` (oppCat ` O ) ) )
31		eqidd 2468	. . . . 5 |- ( ph -> ( Hom f ` S ) = ( Hom f ` S ) )
32		eqidd 2468	. . . . 5 |- ( ph -> (compf ` S ) = (compf ` S ) )
33	6	oppccat 14995	. . . . . 6 |- ( O e. Cat -> (oppCat ` O ) e. Cat )
34	5, 33	syl 16	. . . . 5 |- ( ph -> (oppCat ` O ) e. Cat )
35	7	setccat 15287	. . . . . 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 15221	. . . 4 |- ( ph -> ( C FuncCat S ) = ( (oppCat ` O ) FuncCat S ) )
38	26, 37	syl5eq 2520	. . 3 |- ( ph -> Q = ( (oppCat ` O ) FuncCat S ) )
39	38	oveq2d 6311	. 2 |- ( ph -> ( O Func Q ) = ( O Func ( (oppCat ` O ) FuncCat S ) ) )
40	25, 39	eleqtrrd 2558	1 |- ( ph -> Y e. ( O Func Q ) )

Syntax hints:   -> wi 4   = wceq 1379   e. wcel 1767   C_ wss 3481   X. cxp 5003   dom cdm 5005   ran crn 5006   Rel wrel 5010   Fn wfn 5589   ` cfv 5594  (class class class)co 6295  tpos ctpos 6966   Basecbs 14507   Catccat 14936   Hom _f chomf 14938  comp_fccomf 14939  oppCatcoppc 14984   Func cfunc 15098   FuncCat cfuc 15186   SetCatcsetc 15277  Yoncyon 15393

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-tpos 6967  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-fz 11685  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-hom 14596  df-cco 14597  df-cat 14940  df-cid 14941  df-homf 14942  df-comf 14943  df-oppc 14985  df-func 15102  df-nat 15187  df-fuc 15188  df-setc 15278  df-xpc 15316  df-curf 15358  df-hof 15394  df-yon 15395

This theorem is referenced by:  oyon1cl  15415