Theorem oyoncl 16155

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 15627	. . . 4 |- ( C e. Cat -> O e. Cat )
5	2, 4	syl 17	. . 3 |- ( ph -> O e. Cat )
6		eqid 2451	. . 3 |- (oppCat ` O ) = (oppCat ` O )
7		oyoncl.s	. . 3 |- S = ( SetCat ` U )
8		eqid 2451	. . 3 |- ( (oppCat ` O ) FuncCat S ) = ( (oppCat ` O ) FuncCat S )
9		oyoncl.u	. . 3 |- ( ph -> U e. V )
10		eqid 2451	. . . . . . 7 |- ( Hom f ` C ) = ( Hom f ` C )
11	3, 10	oppchomf 15625	. . . . . 6 |- tpos ( Hom f ` C ) = ( Hom f ` O )
12	11	rneqi 5061	. . . . 5 |- ran tpos ( Hom f ` C ) = ran ( Hom f ` O )
13		relxp 4942	. . . . . . 7 |- Rel ( ( Base ` C ) X. ( Base ` C ) )
14		eqid 2451	. . . . . . . . . 10 |- ( Base ` C ) = ( Base ` C )
15	10, 14	homffn 15598	. . . . . . . . 9 |- ( Hom f ` C ) Fn ( ( Base ` C ) X. ( Base ` C ) )
16		fndm 5675	. . . . . . . . 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 4918	. . . . . . 7 |- ( Rel dom ( Hom f ` C ) <-> Rel ( ( Base ` C ) X. ( Base ` C ) ) )
19	13, 18	mpbir 213	. . . . . 6 |- Rel dom ( Hom f ` C )
20		rntpos 6986	. . . . . 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 2475	. . . 4 |- ran ( Hom f ` O ) = ran ( Hom f ` C )
23		oyoncl.h	. . . 4 |- ( ph -> ran ( Hom f ` C ) C_ U )
24	22, 23	syl5eqss 3476	. . 3 |- ( ph -> ran ( Hom f ` O ) C_ U )
25	1, 5, 6, 7, 8, 9, 24	yoncl 16147	. 2 |- ( ph -> Y e. ( O Func ( (oppCat ` O ) FuncCat S ) ) )
26		oyoncl.q	. . . 4 |- Q = ( C FuncCat S )
27	3	2oppchomf 15629	. . . . . 6 |- ( Hom f ` C ) = ( Hom f ` (oppCat ` O ) )
28	27	a1i 11	. . . . 5 |- ( ph -> ( Hom f ` C ) = ( Hom f ` (oppCat ` O ) ) )
29	3	2oppccomf 15630	. . . . . 6 |- (compf ` C ) = (compf ` (oppCat ` O ) )
30	29	a1i 11	. . . . 5 |- ( ph -> (compf ` C ) = (compf ` (oppCat ` O ) ) )
31		eqidd 2452	. . . . 5 |- ( ph -> ( Hom f ` S ) = ( Hom f ` S ) )
32		eqidd 2452	. . . . 5 |- ( ph -> (compf ` S ) = (compf ` S ) )
33	6	oppccat 15627	. . . . . 6 |- ( O e. Cat -> (oppCat ` O ) e. Cat )
34	5, 33	syl 17	. . . . 5 |- ( ph -> (oppCat ` O ) e. Cat )
35	7	setccat 15980	. . . . . 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 15882	. . . 4 |- ( ph -> ( C FuncCat S ) = ( (oppCat ` O ) FuncCat S ) )
38	26, 37	syl5eq 2497	. . 3 |- ( ph -> Q = ( (oppCat ` O ) FuncCat S ) )
39	38	oveq2d 6306	. 2 |- ( ph -> ( O Func Q ) = ( O Func ( (oppCat ` O ) FuncCat S ) ) )
40	25, 39	eleqtrrd 2532	1 |- ( ph -> Y e. ( O Func Q ) )

Syntax hints:   -> wi 4   = wceq 1444   e. wcel 1887   C_ wss 3404   X. cxp 4832   dom cdm 4834   ran crn 4835   Rel wrel 4839   Fn wfn 5577   ` cfv 5582  (class class class)co 6290  tpos ctpos 6972   Basecbs 15121   Catccat 15570   Hom _f chomf 15572  comp_fccomf 15573  oppCatcoppc 15616   Func cfunc 15759   FuncCat cfuc 15847   SetCatcsetc 15970  Yoncyon 16134

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-tpos 6973  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-ixp 7523  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-fz 11785  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-hom 15214  df-cco 15215  df-cat 15574  df-cid 15575  df-homf 15576  df-comf 15577  df-oppc 15617  df-func 15763  df-nat 15848  df-fuc 15849  df-setc 15971  df-xpc 16057  df-curf 16099  df-hof 16135  df-yon 16136

This theorem is referenced by:  oyon1cl  16156