yon2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > yon2		Structured version Visualization version GIF version

Theorem yon2 18257

Description: Value of the Yoneda embedding at a morphism. (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)
yon12.x	⊢ · = (comp‘𝐶)
yon12.w	⊢ (𝜑 → 𝑊 ∈ 𝐵)
yon2.f	⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑍))
yon2.g	⊢ (𝜑 → 𝐺 ∈ (𝑊𝐻𝑋))

**Assertion**
Ref	Expression
yon2	⊢ (𝜑 → ((((𝑋(2^nd ‘𝑌)𝑍)‘𝐹)‘𝑊)‘𝐺) = (𝐹(⟨𝑊, 𝑋⟩ · 𝑍)𝐺))

**Proof of Theorem yon2**
Step	Hyp	Ref	Expression
1		yon11.y	. . . . . . . . 9 ⊢ 𝑌 = (Yon‘𝐶)
2		yon11.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ Cat)
3		eqid 2728	. . . . . . . . 9 ⊢ (oppCat‘𝐶) = (oppCat‘𝐶)
4		eqid 2728	. . . . . . . . 9 ⊢ (Hom_F‘(oppCat‘𝐶)) = (Hom_F‘(oppCat‘𝐶))
5	1, 2, 3, 4	yonval 18252	. . . . . . . 8 ⊢ (𝜑 → 𝑌 = (⟨𝐶, (oppCat‘𝐶)⟩ curry_F (Hom_F‘(oppCat‘𝐶))))
6	5	fveq2d 6901	. . . . . . 7 ⊢ (𝜑 → (2^nd ‘𝑌) = (2^nd ‘(⟨𝐶, (oppCat‘𝐶)⟩ curry_F (Hom_F‘(oppCat‘𝐶)))))
7	6	oveqd 7437	. . . . . 6 ⊢ (𝜑 → (𝑋(2^nd ‘𝑌)𝑍) = (𝑋(2^nd ‘(⟨𝐶, (oppCat‘𝐶)⟩ curry_F (Hom_F‘(oppCat‘𝐶))))𝑍))
8	7	fveq1d 6899	. . . . 5 ⊢ (𝜑 → ((𝑋(2^nd ‘𝑌)𝑍)‘𝐹) = ((𝑋(2^nd ‘(⟨𝐶, (oppCat‘𝐶)⟩ curry_F (Hom_F‘(oppCat‘𝐶))))𝑍)‘𝐹))
9	8	fveq1d 6899	. . . 4 ⊢ (𝜑 → (((𝑋(2^nd ‘𝑌)𝑍)‘𝐹)‘𝑊) = (((𝑋(2^nd ‘(⟨𝐶, (oppCat‘𝐶)⟩ curry_F (Hom_F‘(oppCat‘𝐶))))𝑍)‘𝐹)‘𝑊))
10		eqid 2728	. . . . 5 ⊢ (⟨𝐶, (oppCat‘𝐶)⟩ curry_F (Hom_F‘(oppCat‘𝐶))) = (⟨𝐶, (oppCat‘𝐶)⟩ curry_F (Hom_F‘(oppCat‘𝐶)))
11		yon11.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
12	3	oppccat 17703	. . . . . 6 ⊢ (𝐶 ∈ Cat → (oppCat‘𝐶) ∈ Cat)
13	2, 12	syl 17	. . . . 5 ⊢ (𝜑 → (oppCat‘𝐶) ∈ Cat)
14		eqid 2728	. . . . . 6 ⊢ (SetCat‘ran (Hom_f ‘𝐶)) = (SetCat‘ran (Hom_f ‘𝐶))
15		fvex 6910	. . . . . . . 8 ⊢ (Hom_f ‘𝐶) ∈ V
16	15	rnex 7918	. . . . . . 7 ⊢ ran (Hom_f ‘𝐶) ∈ V
17	16	a1i 11	. . . . . 6 ⊢ (𝜑 → ran (Hom_f ‘𝐶) ∈ V)
18		ssidd 4003	. . . . . 6 ⊢ (𝜑 → ran (Hom_f ‘𝐶) ⊆ ran (Hom_f ‘𝐶))
19	3, 4, 14, 2, 17, 18	oppchofcl 18251	. . . . 5 ⊢ (𝜑 → (Hom_F‘(oppCat‘𝐶)) ∈ ((𝐶 ×_c (oppCat‘𝐶)) Func (SetCat‘ran (Hom_f ‘𝐶))))
20	3, 11	oppcbas 17698	. . . . 5 ⊢ 𝐵 = (Base‘(oppCat‘𝐶))
21		yon11.h	. . . . 5 ⊢ 𝐻 = (Hom ‘𝐶)
22		eqid 2728	. . . . 5 ⊢ (Id‘(oppCat‘𝐶)) = (Id‘(oppCat‘𝐶))
23		yon11.p	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
24		yon11.z	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
25		yon2.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑍))
26		eqid 2728	. . . . 5 ⊢ ((𝑋(2^nd ‘(⟨𝐶, (oppCat‘𝐶)⟩ curry_F (Hom_F‘(oppCat‘𝐶))))𝑍)‘𝐹) = ((𝑋(2^nd ‘(⟨𝐶, (oppCat‘𝐶)⟩ curry_F (Hom_F‘(oppCat‘𝐶))))𝑍)‘𝐹)
27		yon12.w	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ 𝐵)
28	10, 11, 2, 13, 19, 20, 21, 22, 23, 24, 25, 26, 27	curf2val 18221	. . . 4 ⊢ (𝜑 → (((𝑋(2^nd ‘(⟨𝐶, (oppCat‘𝐶)⟩ curry_F (Hom_F‘(oppCat‘𝐶))))𝑍)‘𝐹)‘𝑊) = (𝐹(⟨𝑋, 𝑊⟩(2^nd ‘(Hom_F‘(oppCat‘𝐶)))⟨𝑍, 𝑊⟩)((Id‘(oppCat‘𝐶))‘𝑊)))
29	9, 28	eqtrd 2768	. . 3 ⊢ (𝜑 → (((𝑋(2^nd ‘𝑌)𝑍)‘𝐹)‘𝑊) = (𝐹(⟨𝑋, 𝑊⟩(2^nd ‘(Hom_F‘(oppCat‘𝐶)))⟨𝑍, 𝑊⟩)((Id‘(oppCat‘𝐶))‘𝑊)))
30	29	fveq1d 6899	. 2 ⊢ (𝜑 → ((((𝑋(2^nd ‘𝑌)𝑍)‘𝐹)‘𝑊)‘𝐺) = ((𝐹(⟨𝑋, 𝑊⟩(2^nd ‘(Hom_F‘(oppCat‘𝐶)))⟨𝑍, 𝑊⟩)((Id‘(oppCat‘𝐶))‘𝑊))‘𝐺))
31		eqid 2728	. . 3 ⊢ (Hom ‘(oppCat‘𝐶)) = (Hom ‘(oppCat‘𝐶))
32		eqid 2728	. . 3 ⊢ (comp‘(oppCat‘𝐶)) = (comp‘(oppCat‘𝐶))
33	21, 3	oppchom 17695	. . . 4 ⊢ (𝑍(Hom ‘(oppCat‘𝐶))𝑋) = (𝑋𝐻𝑍)
34	25, 33	eleqtrrdi 2840	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑍(Hom ‘(oppCat‘𝐶))𝑋))
35	20, 31, 22, 13, 27	catidcl 17661	. . 3 ⊢ (𝜑 → ((Id‘(oppCat‘𝐶))‘𝑊) ∈ (𝑊(Hom ‘(oppCat‘𝐶))𝑊))
36		yon2.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝑊𝐻𝑋))
37	21, 3	oppchom 17695	. . . 4 ⊢ (𝑋(Hom ‘(oppCat‘𝐶))𝑊) = (𝑊𝐻𝑋)
38	36, 37	eleqtrrdi 2840	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝑋(Hom ‘(oppCat‘𝐶))𝑊))
39	4, 13, 20, 31, 23, 27, 24, 27, 32, 34, 35, 38	hof2 18248	. 2 ⊢ (𝜑 → ((𝐹(⟨𝑋, 𝑊⟩(2^nd ‘(Hom_F‘(oppCat‘𝐶)))⟨𝑍, 𝑊⟩)((Id‘(oppCat‘𝐶))‘𝑊))‘𝐺) = ((((Id‘(oppCat‘𝐶))‘𝑊)(⟨𝑋, 𝑊⟩(comp‘(oppCat‘𝐶))𝑊)𝐺)(⟨𝑍, 𝑋⟩(comp‘(oppCat‘𝐶))𝑊)𝐹))
40	20, 31, 22, 13, 23, 32, 27, 38	catlid 17662	. . . 4 ⊢ (𝜑 → (((Id‘(oppCat‘𝐶))‘𝑊)(⟨𝑋, 𝑊⟩(comp‘(oppCat‘𝐶))𝑊)𝐺) = 𝐺)
41	40	oveq1d 7435	. . 3 ⊢ (𝜑 → ((((Id‘(oppCat‘𝐶))‘𝑊)(⟨𝑋, 𝑊⟩(comp‘(oppCat‘𝐶))𝑊)𝐺)(⟨𝑍, 𝑋⟩(comp‘(oppCat‘𝐶))𝑊)𝐹) = (𝐺(⟨𝑍, 𝑋⟩(comp‘(oppCat‘𝐶))𝑊)𝐹))
42		yon12.x	. . . 4 ⊢ · = (comp‘𝐶)
43	11, 42, 3, 24, 23, 27	oppcco 17697	. . 3 ⊢ (𝜑 → (𝐺(⟨𝑍, 𝑋⟩(comp‘(oppCat‘𝐶))𝑊)𝐹) = (𝐹(⟨𝑊, 𝑋⟩ · 𝑍)𝐺))
44	41, 43	eqtrd 2768	. 2 ⊢ (𝜑 → ((((Id‘(oppCat‘𝐶))‘𝑊)(⟨𝑋, 𝑊⟩(comp‘(oppCat‘𝐶))𝑊)𝐺)(⟨𝑍, 𝑋⟩(comp‘(oppCat‘𝐶))𝑊)𝐹) = (𝐹(⟨𝑊, 𝑋⟩ · 𝑍)𝐺))
45	30, 39, 44	3eqtrd 2772	1 ⊢ (𝜑 → ((((𝑋(2^nd ‘𝑌)𝑍)‘𝐹)‘𝑊)‘𝐺) = (𝐹(⟨𝑊, 𝑋⟩ · 𝑍)𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 Vcvv 3471 ⟨cop 4635 ran crn 5679 ‘cfv 6548 (class class class)co 7420 2^nd c2nd 7992 Basecbs 17179 Hom chom 17243 compcco 17244 Catccat 17643 Idccid 17644 Hom_f chomf 17645 oppCatcoppc 17690 SetCatcsetc 18063 curry_F ccurf 18201 Hom_Fchof 18239 Yoncyon 18240

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-tpos 8231 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-er 8724 df-map 8846 df-ixp 8916 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-fz 13517 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-hom 17256 df-cco 17257 df-cat 17647 df-cid 17648 df-homf 17649 df-comf 17650 df-oppc 17691 df-func 17843 df-setc 18064 df-xpc 18162 df-curf 18205 df-hof 18241 df-yon 18242

This theorem is referenced by: yonedalem3b 18270 yonffthlem 18273

W3C validator