Theorem yon12 18432

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

Assertion

Ref	Expression
yon12	⊢ (𝜑 → (((𝑍(2nd ‘((1st ‘𝑌)‘𝑋))𝑊)‘𝐹)‘𝐺) = (𝐺(⟨𝑊, 𝑍⟩ · 𝑋)𝐹))

Proof of Theorem yon12

Step	Hyp	Ref	Expression
1		yon11.y	. . . . . . . . . 10 ⊢ 𝑌 = (Yon‘𝐶)
2		yon11.c	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ Cat)
3		eqid 2761	. . . . . . . . . 10 ⊢ (oppCat‘𝐶) = (oppCat‘𝐶)
4		eqid 2761	. . . . . . . . . 10 ⊢ (HomF‘(oppCat‘𝐶)) = (HomF‘(oppCat‘𝐶))
5	1, 2, 3, 4	yonval 18428	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 = (⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))
6	5	fveq2d 6887	. . . . . . . 8 ⊢ (𝜑 → (1st ‘𝑌) = (1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶)))))
7	6	fveq1d 6885	. . . . . . 7 ⊢ (𝜑 → ((1st ‘𝑌)‘𝑋) = ((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋))
8	7	fveq2d 6887	. . . . . 6 ⊢ (𝜑 → (2nd ‘((1st ‘𝑌)‘𝑋)) = (2nd ‘((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋)))
9	8	oveqd 7435	. . . . 5 ⊢ (𝜑 → (𝑍(2nd ‘((1st ‘𝑌)‘𝑋))𝑊) = (𝑍(2nd ‘((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋))𝑊))
10	9	fveq1d 6885	. . . 4 ⊢ (𝜑 → ((𝑍(2nd ‘((1st ‘𝑌)‘𝑋))𝑊)‘𝐹) = ((𝑍(2nd ‘((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋))𝑊)‘𝐹))
11		eqid 2761	. . . . 5 ⊢ (⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))) = (⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶)))
12		yon11.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
13	3	oppccat 17889	. . . . . 6 ⊢ (𝐶 ∈ Cat → (oppCat‘𝐶) ∈ Cat)
14	2, 13	syl 18	. . . . 5 ⊢ (𝜑 → (oppCat‘𝐶) ∈ Cat)
15		eqid 2761	. . . . . 6 ⊢ (SetCat‘ran (Homf ‘𝐶)) = (SetCat‘ran (Homf ‘𝐶))
16		fvex 6896	. . . . . . . 8 ⊢ (Homf ‘𝐶) ∈ V
17	16	rnex 7920	. . . . . . 7 ⊢ ran (Homf ‘𝐶) ∈ V
18	17	a1i 11	. . . . . 6 ⊢ (𝜑 → ran (Homf ‘𝐶) ∈ V)
19		ssidd 3954	. . . . . 6 ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ ran (Homf ‘𝐶))
20	3, 4, 15, 2, 18, 19	oppchofcl 18427	. . . . 5 ⊢ (𝜑 → (HomF‘(oppCat‘𝐶)) ∈ ((𝐶 ×c (oppCat‘𝐶)) Func (SetCat‘ran (Homf ‘𝐶))))
21	3, 12	oppcbas 17885	. . . . 5 ⊢ 𝐵 = (Base‘(oppCat‘𝐶))
22		yon11.p	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
23		eqid 2761	. . . . 5 ⊢ ((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋) = ((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋)
24		yon11.z	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
25		eqid 2761	. . . . 5 ⊢ (Hom ‘(oppCat‘𝐶)) = (Hom ‘(oppCat‘𝐶))
26		eqid 2761	. . . . 5 ⊢ (Id‘𝐶) = (Id‘𝐶)
27		yon12.w	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ 𝐵)
28		yon12.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝑊𝐻𝑍))
29		yon11.h	. . . . . . 7 ⊢ 𝐻 = (Hom ‘𝐶)
30	29, 3	oppchom 17882	. . . . . 6 ⊢ (𝑍(Hom ‘(oppCat‘𝐶))𝑊) = (𝑊𝐻𝑍)
31	28, 30	eleqtrrdi 2872	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝑍(Hom ‘(oppCat‘𝐶))𝑊))
32	11, 12, 2, 14, 20, 21, 22, 23, 24, 25, 26, 27, 31	curf12 18394	. . . 4 ⊢ (𝜑 → ((𝑍(2nd ‘((1st ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))‘𝑋))𝑊)‘𝐹) = (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑍⟩(2nd ‘(HomF‘(oppCat‘𝐶)))⟨𝑋, 𝑊⟩)𝐹))
33	10, 32	eqtrd 2796	. . 3 ⊢ (𝜑 → ((𝑍(2nd ‘((1st ‘𝑌)‘𝑋))𝑊)‘𝐹) = (((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑍⟩(2nd ‘(HomF‘(oppCat‘𝐶)))⟨𝑋, 𝑊⟩)𝐹))
34	33	fveq1d 6885	. 2 ⊢ (𝜑 → (((𝑍(2nd ‘((1st ‘𝑌)‘𝑋))𝑊)‘𝐹)‘𝐺) = ((((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑍⟩(2nd ‘(HomF‘(oppCat‘𝐶)))⟨𝑋, 𝑊⟩)𝐹)‘𝐺))
35		eqid 2761	. . 3 ⊢ (comp‘(oppCat‘𝐶)) = (comp‘(oppCat‘𝐶))
36	12, 29, 26, 2, 22	catidcl 17849	. . . 4 ⊢ (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋𝐻𝑋))
37	29, 3	oppchom 17882	. . . 4 ⊢ (𝑋(Hom ‘(oppCat‘𝐶))𝑋) = (𝑋𝐻𝑋)
38	36, 37	eleqtrrdi 2872	. . 3 ⊢ (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋(Hom ‘(oppCat‘𝐶))𝑋))
39		yon12.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝑍𝐻𝑋))
40	29, 3	oppchom 17882	. . . 4 ⊢ (𝑋(Hom ‘(oppCat‘𝐶))𝑍) = (𝑍𝐻𝑋)
41	39, 40	eleqtrrdi 2872	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝑋(Hom ‘(oppCat‘𝐶))𝑍))
42	4, 14, 21, 25, 22, 24, 22, 27, 35, 38, 31, 41	hof2 18424	. 2 ⊢ (𝜑 → ((((Id‘𝐶)‘𝑋)(⟨𝑋, 𝑍⟩(2nd ‘(HomF‘(oppCat‘𝐶)))⟨𝑋, 𝑊⟩)𝐹)‘𝐺) = ((𝐹(⟨𝑋, 𝑍⟩(comp‘(oppCat‘𝐶))𝑊)𝐺)(⟨𝑋, 𝑋⟩(comp‘(oppCat‘𝐶))𝑊)((Id‘𝐶)‘𝑋)))
43		yon12.x	. . . . 5 ⊢ · = (comp‘𝐶)
44	12, 43, 3, 22, 24, 27	oppcco 17884	. . . 4 ⊢ (𝜑 → (𝐹(⟨𝑋, 𝑍⟩(comp‘(oppCat‘𝐶))𝑊)𝐺) = (𝐺(⟨𝑊, 𝑍⟩ · 𝑋)𝐹))
45	44	oveq1d 7433	. . 3 ⊢ (𝜑 → ((𝐹(⟨𝑋, 𝑍⟩(comp‘(oppCat‘𝐶))𝑊)𝐺)(⟨𝑋, 𝑋⟩(comp‘(oppCat‘𝐶))𝑊)((Id‘𝐶)‘𝑋)) = ((𝐺(⟨𝑊, 𝑍⟩ · 𝑋)𝐹)(⟨𝑋, 𝑋⟩(comp‘(oppCat‘𝐶))𝑊)((Id‘𝐶)‘𝑋)))
46	12, 43, 3, 22, 22, 27	oppcco 17884	. . 3 ⊢ (𝜑 → ((𝐺(⟨𝑊, 𝑍⟩ · 𝑋)𝐹)(⟨𝑋, 𝑋⟩(comp‘(oppCat‘𝐶))𝑊)((Id‘𝐶)‘𝑋)) = (((Id‘𝐶)‘𝑋)(⟨𝑊, 𝑋⟩ · 𝑋)(𝐺(⟨𝑊, 𝑍⟩ · 𝑋)𝐹)))
47	12, 29, 43, 2, 27, 24, 22, 28, 39	catcocl 17852	. . . 4 ⊢ (𝜑 → (𝐺(⟨𝑊, 𝑍⟩ · 𝑋)𝐹) ∈ (𝑊𝐻𝑋))
48	12, 29, 26, 2, 27, 43, 22, 47	catlid 17850	. . 3 ⊢ (𝜑 → (((Id‘𝐶)‘𝑋)(⟨𝑊, 𝑋⟩ · 𝑋)(𝐺(⟨𝑊, 𝑍⟩ · 𝑋)𝐹)) = (𝐺(⟨𝑊, 𝑍⟩ · 𝑋)𝐹))
49	45, 46, 48	3eqtrd 2800	. 2 ⊢ (𝜑 → ((𝐹(⟨𝑋, 𝑍⟩(comp‘(oppCat‘𝐶))𝑊)𝐺)(⟨𝑋, 𝑋⟩(comp‘(oppCat‘𝐶))𝑊)((Id‘𝐶)‘𝑋)) = (𝐺(⟨𝑊, 𝑍⟩ · 𝑋)𝐹))
50	34, 42, 49	3eqtrd 2800	1 ⊢ (𝜑 → (((𝑍(2nd ‘((1st ‘𝑌)‘𝑋))𝑊)‘𝐹)‘𝐺) = (𝐺(⟨𝑊, 𝑍⟩ · 𝑋)𝐹))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  Vcvv 3451  ⟨cop 4590  ran crn 5652  ‘cfv 6537  (class class class)co 7418  1^st c1st 7997  2^nd c2nd 7998  Basecbs 17380  Hom chom 17432  compcco 17433  Catccat 17831  Idccid 17832  Hom_f chomf 17833  oppCatcoppc 17878  SetCatcsetc 18243   curry_F ccurf 18377  Hom_Fchof 18415  Yoncyon 18416

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-cnex 11249  ax-resscn 11250  ax-1cn 11251  ax-icn 11252  ax-addcl 11253  ax-addrcl 11254  ax-mulcl 11255  ax-mulrcl 11256  ax-mulcom 11257  ax-addass 11258  ax-mulass 11259  ax-distr 11260  ax-i2m1 11261  ax-1ne0 11262  ax-1rid 11263  ax-rnegex 11264  ax-rrecex 11265  ax-cnre 11266  ax-pre-lttri 11267  ax-pre-lttrn 11268  ax-pre-ltadd 11269  ax-pre-mulgt0 11270

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-1st 7999  df-2nd 8000  df-tpos 8236  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-1o 8469  df-er 8710  df-map 8842  df-ixp 8919  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-pnf 11338  df-mnf 11339  df-xr 11340  df-ltxr 11341  df-le 11342  df-sub 11536  df-neg 11537  df-nn 12329  df-2 12398  df-3 12399  df-4 12400  df-5 12401  df-6 12402  df-7 12403  df-8 12404  df-9 12405  df-n0 12600  df-z 12687  df-dec 12808  df-uz 12959  df-fz 13633  df-struct 17318  df-sets 17335  df-slot 17353  df-ndx 17365  df-base 17381  df-hom 17445  df-cco 17446  df-cat 17835  df-cid 17836  df-homf 17837  df-comf 17838  df-oppc 17879  df-func 18026  df-setc 18244  df-xpc 18339  df-curf 18381  df-hof 18417  df-yon 18418

This theorem is used by:  yonedalem4c  18444  yonedalem3b  18446  yonedainv  18448  yonffthlem  18449