Theorem yon11 18059

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 18056	. . . . . 6 ⊢ (𝜑 → 𝑌 = (⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))
6	5	fveq2d 6816	. . . . 5 ⊢ (𝜑 → (1st ‘𝑌) = (1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶)))))
7	6	fveq1d 6814	. . . 4 ⊢ (𝜑 → ((1st ‘𝑌)‘𝑋) = ((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋))
8	7	fveq2d 6816	. . 3 ⊢ (𝜑 → (1st ‘((1st ‘𝑌)‘𝑋)) = (1st ‘((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋)))
9	8	fveq1d 6814	. 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 17510	. . . 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 6825	. . . . . 6 ⊢ (Homf ‘𝐶) ∈ V
16	15	rnex 7806	. . . . 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 18055	. . 3 ⊢ (𝜑 → (HomF‘(oppCat‘𝐶)) ∈ ((𝐶 ×c (oppCat‘𝐶)) Func (SetCat‘ran (Homf ‘𝐶))))
20	3, 11	oppcbas 17505	. . 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 18021	. 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 18049	. . 3 ⊢ (𝜑 → (𝑋(1st ‘(HomF‘(oppCat‘𝐶)))𝑍) = (𝑋(Hom ‘(oppCat‘𝐶))𝑍))
27		yon11.h	. . . 4 ⊢ 𝐻 = (Hom ‘𝐶)
28	27, 3	oppchom 17502	. . 3 ⊢ (𝑋(Hom ‘(oppCat‘𝐶))𝑍) = (𝑍𝐻𝑋)
29	26, 28	eqtrdi 2793	. 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 1540   ∈ wcel 2105  Vcvv 3441  ⟨cop 4577  ran crn 5609  ‘cfv 6466  (class class class)co 7317  1^st c1st 7876  Basecbs 16989  Hom chom 17050  Catccat 17450  Hom_f chomf 17452  oppCatcoppc 17497  SetCatcsetc 17867   curry_F ccurf 18005  Hom_Fchof 18043  Yoncyon 18044

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-rep 5224  ax-sep 5238  ax-nul 5245  ax-pow 5303  ax-pr 5367  ax-un 7630  ax-cnex 11007  ax-resscn 11008  ax-1cn 11009  ax-icn 11010  ax-addcl 11011  ax-addrcl 11012  ax-mulcl 11013  ax-mulrcl 11014  ax-mulcom 11015  ax-addass 11016  ax-mulass 11017  ax-distr 11018  ax-i2m1 11019  ax-1ne0 11020  ax-1rid 11021  ax-rnegex 11022  ax-rrecex 11023  ax-cnre 11024  ax-pre-lttri 11025  ax-pre-lttrn 11026  ax-pre-ltadd 11027  ax-pre-mulgt0 11028

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3350  df-reu 3351  df-rab 3405  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4268  df-if 4472  df-pw 4547  df-sn 4572  df-pr 4574  df-tp 4576  df-op 4578  df-uni 4851  df-iun 4939  df-br 5088  df-opab 5150  df-mpt 5171  df-tr 5205  df-id 5507  df-eprel 5513  df-po 5521  df-so 5522  df-fr 5563  df-we 5565  df-xp 5614  df-rel 5615  df-cnv 5616  df-co 5617  df-dm 5618  df-rn 5619  df-res 5620  df-ima 5621  df-pred 6225  df-ord 6292  df-on 6293  df-lim 6294  df-suc 6295  df-iota 6418  df-fun 6468  df-fn 6469  df-f 6470  df-f1 6471  df-fo 6472  df-f1o 6473  df-fv 6474  df-riota 7274  df-ov 7320  df-oprab 7321  df-mpo 7322  df-om 7760  df-1st 7878  df-2nd 7879  df-tpos 8091  df-frecs 8146  df-wrecs 8177  df-recs 8251  df-rdg 8290  df-1o 8346  df-er 8548  df-map 8667  df-ixp 8736  df-en 8784  df-dom 8785  df-sdom 8786  df-fin 8787  df-pnf 11091  df-mnf 11092  df-xr 11093  df-ltxr 11094  df-le 11095  df-sub 11287  df-neg 11288  df-nn 12054  df-2 12116  df-3 12117  df-4 12118  df-5 12119  df-6 12120  df-7 12121  df-8 12122  df-9 12123  df-n0 12314  df-z 12400  df-dec 12518  df-uz 12663  df-fz 13320  df-struct 16925  df-sets 16942  df-slot 16960  df-ndx 16972  df-base 16990  df-hom 17063  df-cco 17064  df-cat 17454  df-cid 17455  df-homf 17456  df-comf 17457  df-oppc 17498  df-func 17650  df-setc 17868  df-xpc 17966  df-curf 18009  df-hof 18045  df-yon 18046

This theorem is referenced by:  yonedalem3a  18069  yonedalem4c  18072  yonedalem3b  18074  yonedainv  18076  yonffthlem  18077  yoniso  18080