Theorem oyoncl 15182

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 14763	. . . 4 |- ( C e. Cat -> O e. Cat )
5	2, 4	syl 16	. . 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 14761	. . . . . 6 |- tpos ( Hom f ` C ) = ( Hom f ` O )
12	11	rneqi 5164	. . . . 5 |- ran tpos ( Hom f ` C ) = ran ( Hom f ` O )
13		relxp 5045	. . . . . . 7 |- Rel ( ( Base ` C ) X. ( Base ` C ) )
14		eqid 2451	. . . . . . . . . 10 |- ( Base ` C ) = ( Base ` C )
15	10, 14	homffn 14734	. . . . . . . . 9 |- ( Hom f ` C ) Fn ( ( Base ` C ) X. ( Base ` C ) )
16		fndm 5608	. . . . . . . . 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 5021	. . . . . . 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 6858	. . . . . 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 2482	. . . 4 |- ran ( Hom f ` O ) = ran ( Hom f ` C )
23		oyoncl.h	. . . 4 |- ( ph -> ran ( Hom f ` C ) C_ U )
24	22, 23	syl5eqss 3498	. . 3 |- ( ph -> ran ( Hom f ` O ) C_ U )
25	1, 5, 6, 7, 8, 9, 24	yoncl 15174	. 2 |- ( ph -> Y e. ( O Func ( (oppCat ` O ) FuncCat S ) ) )
26		oyoncl.q	. . . 4 |- Q = ( C FuncCat S )
27	3	2oppchomf 14765	. . . . . 6 |- ( Hom f ` C ) = ( Hom f ` (oppCat ` O ) )
28	27	a1i 11	. . . . 5 |- ( ph -> ( Hom f ` C ) = ( Hom f ` (oppCat ` O ) ) )
29	3	2oppccomf 14766	. . . . . 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 14763	. . . . . 6 |- ( O e. Cat -> (oppCat ` O ) e. Cat )
34	5, 33	syl 16	. . . . 5 |- ( ph -> (oppCat ` O ) e. Cat )
35	7	setccat 15055	. . . . . 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 14989	. . . 4 |- ( ph -> ( C FuncCat S ) = ( (oppCat ` O ) FuncCat S ) )
38	26, 37	syl5eq 2504	. . 3 |- ( ph -> Q = ( (oppCat ` O ) FuncCat S ) )
39	38	oveq2d 6206	. 2 |- ( ph -> ( O Func Q ) = ( O Func ( (oppCat ` O ) FuncCat S ) ) )
40	25, 39	eleqtrrd 2542	1 |- ( ph -> Y e. ( O Func Q ) )

Syntax hints:   -> wi 4   = wceq 1370   e. wcel 1758   C_ wss 3426   X. cxp 4936   dom cdm 4938   ran crn 4939   Rel wrel 4943   Fn wfn 5511   ` cfv 5516  (class class class)co 6190  tpos ctpos 6844   Basecbs 14276   Catccat 14704   Hom _f chomf 14706  comp_fccomf 14707  oppCatcoppc 14752   Func cfunc 14866   FuncCat cfuc 14954   SetCatcsetc 15045  Yoncyon 15161

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-1st 6677  df-2nd 6678  df-tpos 6845  df-recs 6932  df-rdg 6966  df-1o 7020  df-oadd 7024  df-er 7201  df-map 7316  df-ixp 7364  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-nn 10424  df-2 10481  df-3 10482  df-4 10483  df-5 10484  df-6 10485  df-7 10486  df-8 10487  df-9 10488  df-10 10489  df-n0 10681  df-z 10748  df-dec 10857  df-uz 10963  df-fz 11539  df-struct 14278  df-ndx 14279  df-slot 14280  df-base 14281  df-sets 14282  df-hom 14364  df-cco 14365  df-cat 14708  df-cid 14709  df-homf 14710  df-comf 14711  df-oppc 14753  df-func 14870  df-nat 14955  df-fuc 14956  df-setc 15046  df-xpc 15084  df-curf 15126  df-hof 15162  df-yon 15163

This theorem is referenced by:  oyon1cl  15183