Theorem yon11 17772

Description: Value of the Yoneda embedding at an object. The partially evaluated Yoneda embedding is also the contravariant Hom functor. (Contributed by Mario Carneiro, 17-Jan-2017.)

Hypotheses

Ref	Expression
yon11.y	⊢ 𝑌 = (Yon‘𝐶)
yon11.b	⊢ 𝐵 = (Base‘𝐶)
yon11.c	⊢ (𝜑 → 𝐶 ∈ Cat)
yon11.p	⊢ (𝜑 → 𝑋 ∈ 𝐵)
yon11.h	⊢ 𝐻 = (Hom ‘𝐶)
yon11.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)

Assertion

Ref	Expression
yon11	⊢ (𝜑 → ((1st ‘((1st ‘𝑌)‘𝑋))‘𝑍) = (𝑍𝐻𝑋))

Proof of Theorem yon11

Step	Hyp	Ref	Expression
1		yon11.y	. . . . . . 7 ⊢ 𝑌 = (Yon‘𝐶)
2		yon11.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ Cat)
3		eqid 2737	. . . . . . 7 ⊢ (oppCat‘𝐶) = (oppCat‘𝐶)
4		eqid 2737	. . . . . . 7 ⊢ (HomF‘(oppCat‘𝐶)) = (HomF‘(oppCat‘𝐶))
5	1, 2, 3, 4	yonval 17769	. . . . . 6 ⊢ (𝜑 → 𝑌 = (⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))
6	5	fveq2d 6721	. . . . 5 ⊢ (𝜑 → (1st ‘𝑌) = (1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶)))))
7	6	fveq1d 6719	. . . 4 ⊢ (𝜑 → ((1st ‘𝑌)‘𝑋) = ((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋))
8	7	fveq2d 6721	. . 3 ⊢ (𝜑 → (1st ‘((1st ‘𝑌)‘𝑋)) = (1st ‘((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋)))
9	8	fveq1d 6719	. 2 ⊢ (𝜑 → ((1st ‘((1st ‘𝑌)‘𝑋))‘𝑍) = ((1st ‘((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋))‘𝑍))
10		eqid 2737	. . 3 ⊢ (⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))) = (⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶)))
11		yon11.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
12	3	oppccat 17226	. . . 4 ⊢ (𝐶 ∈ Cat → (oppCat‘𝐶) ∈ Cat)
13	2, 12	syl 17	. . 3 ⊢ (𝜑 → (oppCat‘𝐶) ∈ Cat)
14		eqid 2737	. . . 4 ⊢ (SetCat‘ran (Homf ‘𝐶)) = (SetCat‘ran (Homf ‘𝐶))
15		fvex 6730	. . . . . 6 ⊢ (Homf ‘𝐶) ∈ V
16	15	rnex 7690	. . . . 5 ⊢ ran (Homf ‘𝐶) ∈ V
17	16	a1i 11	. . . 4 ⊢ (𝜑 → ran (Homf ‘𝐶) ∈ V)
18		ssidd 3924	. . . 4 ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ ran (Homf ‘𝐶))
19	3, 4, 14, 2, 17, 18	oppchofcl 17768	. . 3 ⊢ (𝜑 → (HomF‘(oppCat‘𝐶)) ∈ ((𝐶 ×c (oppCat‘𝐶)) Func (SetCat‘ran (Homf ‘𝐶))))
20	3, 11	oppcbas 17222	. . 3 ⊢ 𝐵 = (Base‘(oppCat‘𝐶))
21		yon11.p	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
22		eqid 2737	. . 3 ⊢ ((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋) = ((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋)
23		yon11.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
24	10, 11, 2, 13, 19, 20, 21, 22, 23	curf11 17734	. 2 ⊢ (𝜑 → ((1st ‘((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋))‘𝑍) = (𝑋(1st ‘(HomF‘(oppCat‘𝐶)))𝑍))
25		eqid 2737	. . . 4 ⊢ (Hom ‘(oppCat‘𝐶)) = (Hom ‘(oppCat‘𝐶))
26	4, 13, 20, 25, 21, 23	hof1 17762	. . 3 ⊢ (𝜑 → (𝑋(1st ‘(HomF‘(oppCat‘𝐶)))𝑍) = (𝑋(Hom ‘(oppCat‘𝐶))𝑍))
27		yon11.h	. . . 4 ⊢ 𝐻 = (Hom ‘𝐶)
28	27, 3	oppchom 17219	. . 3 ⊢ (𝑋(Hom ‘(oppCat‘𝐶))𝑍) = (𝑍𝐻𝑋)
29	26, 28	eqtrdi 2794	. 2 ⊢ (𝜑 → (𝑋(1st ‘(HomF‘(oppCat‘𝐶)))𝑍) = (𝑍𝐻𝑋))
30	9, 24, 29	3eqtrd 2781	1 ⊢ (𝜑 → ((1st ‘((1st ‘𝑌)‘𝑋))‘𝑍) = (𝑍𝐻𝑋))

Syntax hints:   → wi 4   = wceq 1543   ∈ wcel 2110  Vcvv 3408  ⟨cop 4547  ran crn 5552  ‘cfv 6380  (class class class)co 7213  1^st c1st 7759  Basecbs 16760  Hom chom 16813  Catccat 17167  Hom_f chomf 17169  oppCatcoppc 17214  SetCatcsetc 17581   curry_F ccurf 17718  Hom_Fchof 17756  Yoncyon 17757

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-tpos 7968  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-er 8391  df-map 8510  df-ixp 8579  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-nn 11831  df-2 11893  df-3 11894  df-4 11895  df-5 11896  df-6 11897  df-7 11898  df-8 11899  df-9 11900  df-n0 12091  df-z 12177  df-dec 12294  df-uz 12439  df-fz 13096  df-struct 16700  df-sets 16717  df-slot 16735  df-ndx 16745  df-base 16761  df-hom 16826  df-cco 16827  df-cat 17171  df-cid 17172  df-homf 17173  df-comf 17174  df-oppc 17215  df-func 17364  df-setc 17582  df-xpc 17679  df-curf 17722  df-hof 17758  df-yon 17759

This theorem is referenced by:  yonedalem3a  17782  yonedalem4c  17785  yonedalem3b  17787  yonedainv  17789  yonffthlem  17790  yoniso  17793