Theorem yon2 18279

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(2nd ‘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 2761	. . . . . . . . 9 ⊢ (oppCat‘𝐶) = (oppCat‘𝐶)
4		eqid 2761	. . . . . . . . 9 ⊢ (HomF‘(oppCat‘𝐶)) = (HomF‘(oppCat‘𝐶))
5	1, 2, 3, 4	yonval 18274	. . . . . . . 8 ⊢ (𝜑 → 𝑌 = (⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))
6	5	fveq2d 6865	. . . . . . 7 ⊢ (𝜑 → (2nd ‘𝑌) = (2nd ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶)))))
7	6	oveqd 7407	. . . . . 6 ⊢ (𝜑 → (𝑋(2nd ‘𝑌)𝑍) = (𝑋(2nd ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))𝑍))
8	7	fveq1d 6863	. . . . 5 ⊢ (𝜑 → ((𝑋(2nd ‘𝑌)𝑍)‘𝐹) = ((𝑋(2nd ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))𝑍)‘𝐹))
9	8	fveq1d 6863	. . . 4 ⊢ (𝜑 → (((𝑋(2nd ‘𝑌)𝑍)‘𝐹)‘𝑊) = (((𝑋(2nd ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))𝑍)‘𝐹)‘𝑊))
10		eqid 2761	. . . . 5 ⊢ (⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))) = (⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶)))
11		yon11.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
12	3	oppccat 17735	. . . . . 6 ⊢ (𝐶 ∈ Cat → (oppCat‘𝐶) ∈ Cat)
13	2, 12	syl 17	. . . . 5 ⊢ (𝜑 → (oppCat‘𝐶) ∈ Cat)
14		eqid 2761	. . . . . 6 ⊢ (SetCat‘ran (Homf ‘𝐶)) = (SetCat‘ran (Homf ‘𝐶))
15		fvex 6874	. . . . . . . 8 ⊢ (Homf ‘𝐶) ∈ V
16	15	rnex 7885	. . . . . . 7 ⊢ ran (Homf ‘𝐶) ∈ V
17	16	a1i 11	. . . . . 6 ⊢ (𝜑 → ran (Homf ‘𝐶) ∈ V)
18		ssidd 3959	. . . . . 6 ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ ran (Homf ‘𝐶))
19	3, 4, 14, 2, 17, 18	oppchofcl 18273	. . . . 5 ⊢ (𝜑 → (HomF‘(oppCat‘𝐶)) ∈ ((𝐶 ×c (oppCat‘𝐶)) Func (SetCat‘ran (Homf ‘𝐶))))
20	3, 11	oppcbas 17731	. . . . 5 ⊢ 𝐵 = (Base‘(oppCat‘𝐶))
21		yon11.h	. . . . 5 ⊢ 𝐻 = (Hom ‘𝐶)
22		eqid 2761	. . . . 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 2761	. . . . 5 ⊢ ((𝑋(2nd ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))𝑍)‘𝐹) = ((𝑋(2nd ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(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 18243	. . . 4 ⊢ (𝜑 → (((𝑋(2nd ‘(⟨𝐶, (oppCat‘𝐶)⟩ curryF (HomF‘(oppCat‘𝐶))))𝑍)‘𝐹)‘𝑊) = (𝐹(⟨𝑋, 𝑊⟩(2nd ‘(HomF‘(oppCat‘𝐶)))⟨𝑍, 𝑊⟩)((Id‘(oppCat‘𝐶))‘𝑊)))
29	9, 28	eqtrd 2796	. . 3 ⊢ (𝜑 → (((𝑋(2nd ‘𝑌)𝑍)‘𝐹)‘𝑊) = (𝐹(⟨𝑋, 𝑊⟩(2nd ‘(HomF‘(oppCat‘𝐶)))⟨𝑍, 𝑊⟩)((Id‘(oppCat‘𝐶))‘𝑊)))
30	29	fveq1d 6863	. 2 ⊢ (𝜑 → ((((𝑋(2nd ‘𝑌)𝑍)‘𝐹)‘𝑊)‘𝐺) = ((𝐹(⟨𝑋, 𝑊⟩(2nd ‘(HomF‘(oppCat‘𝐶)))⟨𝑍, 𝑊⟩)((Id‘(oppCat‘𝐶))‘𝑊))‘𝐺))
31		eqid 2761	. . 3 ⊢ (Hom ‘(oppCat‘𝐶)) = (Hom ‘(oppCat‘𝐶))
32		eqid 2761	. . 3 ⊢ (comp‘(oppCat‘𝐶)) = (comp‘(oppCat‘𝐶))
33	21, 3	oppchom 17728	. . . 4 ⊢ (𝑍(Hom ‘(oppCat‘𝐶))𝑋) = (𝑋𝐻𝑍)
34	25, 33	eleqtrrdi 2872	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑍(Hom ‘(oppCat‘𝐶))𝑋))
35	20, 31, 22, 13, 27	catidcl 17695	. . 3 ⊢ (𝜑 → ((Id‘(oppCat‘𝐶))‘𝑊) ∈ (𝑊(Hom ‘(oppCat‘𝐶))𝑊))
36		yon2.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝑊𝐻𝑋))
37	21, 3	oppchom 17728	. . . 4 ⊢ (𝑋(Hom ‘(oppCat‘𝐶))𝑊) = (𝑊𝐻𝑋)
38	36, 37	eleqtrrdi 2872	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝑋(Hom ‘(oppCat‘𝐶))𝑊))
39	4, 13, 20, 31, 23, 27, 24, 27, 32, 34, 35, 38	hof2 18270	. 2 ⊢ (𝜑 → ((𝐹(⟨𝑋, 𝑊⟩(2nd ‘(HomF‘(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 17696	. . . 4 ⊢ (𝜑 → (((Id‘(oppCat‘𝐶))‘𝑊)(⟨𝑋, 𝑊⟩(comp‘(oppCat‘𝐶))𝑊)𝐺) = 𝐺)
41	40	oveq1d 7405	. . 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 17730	. . 3 ⊢ (𝜑 → (𝐺(⟨𝑍, 𝑋⟩(comp‘(oppCat‘𝐶))𝑊)𝐹) = (𝐹(⟨𝑊, 𝑋⟩ · 𝑍)𝐺))
44	41, 43	eqtrd 2796	. 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 2800	1 ⊢ (𝜑 → ((((𝑋(2nd ‘𝑌)𝑍)‘𝐹)‘𝑊)‘𝐺) = (𝐹(⟨𝑊, 𝑋⟩ · 𝑍)𝐺))

Syntax hints:   → wi 4   = wceq 1559   ∈ wcel 2141  Vcvv 3453  ⟨cop 4587  ran crn 5646  ‘cfv 6515  (class class class)co 7390  2^nd c2nd 7963  Basecbs 17226  Hom chom 17278  compcco 17279  Catccat 17677  Idccid 17678  Hom_f chomf 17679  oppCatcoppc 17724  SetCatcsetc 18089   curry_F ccurf 18223  Hom_Fchof 18261  Yoncyon 18262

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7712  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6282  df-ord 6343  df-on 6344  df-lim 6345  df-suc 6346  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522  df-fv 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7841  df-1st 7964  df-2nd 7965  df-tpos 8199  df-frecs 8255  df-wrecs 8286  df-recs 8335  df-rdg 8374  df-1o 8430  df-er 8671  df-map 8803  df-ixp 8874  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-pnf 11213  df-mnf 11214  df-xr 11215  df-ltxr 11216  df-le 11217  df-sub 11411  df-neg 11412  df-nn 12206  df-2 12275  df-3 12276  df-4 12277  df-5 12278  df-6 12279  df-7 12280  df-8 12281  df-9 12282  df-n0 12477  df-z 12564  df-dec 12684  df-uz 12835  df-fz 13508  df-struct 17164  df-sets 17181  df-slot 17199  df-ndx 17211  df-base 17227  df-hom 17291  df-cco 17292  df-cat 17681  df-cid 17682  df-homf 17683  df-comf 17684  df-oppc 17725  df-func 17872  df-setc 18090  df-xpc 18185  df-curf 18227  df-hof 18263  df-yon 18264

This theorem is referenced by:  yonedalem3b  18292  yonffthlem  18295