Theorem yon11 18178

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 2733	. . . . . . 7 ⊢ (oppCat‘𝐶) = (oppCat‘𝐶)
4		eqid 2733	. . . . . . 7 ⊢ (HomF‘(oppCat‘𝐶)) = (HomF‘(oppCat‘𝐶))
5	1, 2, 3, 4	yonval 18175	. . . . . 6 ⊢ (𝜑 → 𝑌 = (⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))
6	5	fveq2d 6835	. . . . 5 ⊢ (𝜑 → (1st ‘𝑌) = (1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶)))))
7	6	fveq1d 6833	. . . 4 ⊢ (𝜑 → ((1st ‘𝑌)‘𝑋) = ((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋))
8	7	fveq2d 6835	. . 3 ⊢ (𝜑 → (1st ‘((1st ‘𝑌)‘𝑋)) = (1st ‘((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋)))
9	8	fveq1d 6833	. 2 ⊢ (𝜑 → ((1st ‘((1st ‘𝑌)‘𝑋))‘𝑍) = ((1st ‘((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋))‘𝑍))
10		eqid 2733	. . 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 17636	. . . 4 ⊢ (𝐶 ∈ Cat → (oppCat‘𝐶) ∈ Cat)
13	2, 12	syl 17	. . 3 ⊢ (𝜑 → (oppCat‘𝐶) ∈ Cat)
14		eqid 2733	. . . 4 ⊢ (SetCat‘ran (Homf ‘𝐶)) = (SetCat‘ran (Homf ‘𝐶))
15		fvex 6844	. . . . . 6 ⊢ (Homf ‘𝐶) ∈ V
16	15	rnex 7849	. . . . 5 ⊢ ran (Homf ‘𝐶) ∈ V
17	16	a1i 11	. . . 4 ⊢ (𝜑 → ran (Homf ‘𝐶) ∈ V)
18		ssidd 3954	. . . 4 ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ ran (Homf ‘𝐶))
19	3, 4, 14, 2, 17, 18	oppchofcl 18174	. . 3 ⊢ (𝜑 → (HomF‘(oppCat‘𝐶)) ∈ ((𝐶 ×c (oppCat‘𝐶)) Func (SetCat‘ran (Homf ‘𝐶))))
20	3, 11	oppcbas 17632	. . 3 ⊢ 𝐵 = (Base‘(oppCat‘𝐶))
21		yon11.p	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
22		eqid 2733	. . 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 18140	. 2 ⊢ (𝜑 → ((1st ‘((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋))‘𝑍) = (𝑋(1st ‘(HomF‘(oppCat‘𝐶)))𝑍))
25		eqid 2733	. . . 4 ⊢ (Hom ‘(oppCat‘𝐶)) = (Hom ‘(oppCat‘𝐶))
26	4, 13, 20, 25, 21, 23	hof1 18168	. . 3 ⊢ (𝜑 → (𝑋(1st ‘(HomF‘(oppCat‘𝐶)))𝑍) = (𝑋(Hom ‘(oppCat‘𝐶))𝑍))
27		yon11.h	. . . 4 ⊢ 𝐻 = (Hom ‘𝐶)
28	27, 3	oppchom 17629	. . 3 ⊢ (𝑋(Hom ‘(oppCat‘𝐶))𝑍) = (𝑍𝐻𝑋)
29	26, 28	eqtrdi 2784	. 2 ⊢ (𝜑 → (𝑋(1st ‘(HomF‘(oppCat‘𝐶)))𝑍) = (𝑍𝐻𝑋))
30	9, 24, 29	3eqtrd 2772	1 ⊢ (𝜑 → ((1st ‘((1st ‘𝑌)‘𝑋))‘𝑍) = (𝑍𝐻𝑋))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  Vcvv 3437  ⟨cop 4583  ran crn 5622  ‘cfv 6489  (class class class)co 7355  1^st c1st 7928  Basecbs 17127  Hom chom 17179  Catccat 17578  Hom_f chomf 17580  oppCatcoppc 17625  SetCatcsetc 17990   curry_F ccurf 18124  Hom_Fchof 18162  Yoncyon 18163

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677  ax-cnex 11073  ax-resscn 11074  ax-1cn 11075  ax-icn 11076  ax-addcl 11077  ax-addrcl 11078  ax-mulcl 11079  ax-mulrcl 11080  ax-mulcom 11081  ax-addass 11082  ax-mulass 11083  ax-distr 11084  ax-i2m1 11085  ax-1ne0 11086  ax-1rid 11087  ax-rnegex 11088  ax-rrecex 11089  ax-cnre 11090  ax-pre-lttri 11091  ax-pre-lttrn 11092  ax-pre-ltadd 11093  ax-pre-mulgt0 11094

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7312  df-ov 7358  df-oprab 7359  df-mpo 7360  df-om 7806  df-1st 7930  df-2nd 7931  df-tpos 8165  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-1o 8394  df-er 8631  df-map 8761  df-ixp 8832  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-pnf 11159  df-mnf 11160  df-xr 11161  df-ltxr 11162  df-le 11163  df-sub 11357  df-neg 11358  df-nn 12137  df-2 12199  df-3 12200  df-4 12201  df-5 12202  df-6 12203  df-7 12204  df-8 12205  df-9 12206  df-n0 12393  df-z 12480  df-dec 12599  df-uz 12743  df-fz 13415  df-struct 17065  df-sets 17082  df-slot 17100  df-ndx 17112  df-base 17128  df-hom 17192  df-cco 17193  df-cat 17582  df-cid 17583  df-homf 17584  df-comf 17585  df-oppc 17626  df-func 17773  df-setc 17991  df-xpc 18086  df-curf 18128  df-hof 18164  df-yon 18165

This theorem is referenced by:  yonedalem3a  18188  yonedalem4c  18191  yonedalem3b  18193  yonedainv  18195  yonffthlem  18196  yoniso  18199