Theorem yonedalem4a 18230

Description: Lemma for yoneda 18238. (Contributed by Mario Carneiro, 29-Jan-2017.)

Hypotheses

Ref	Expression
yoneda.y	⊢ 𝑌 = (Yon‘𝐶)
yoneda.b	⊢ 𝐵 = (Base‘𝐶)
yoneda.1	⊢ 1 = (Id‘𝐶)
yoneda.o	⊢ 𝑂 = (oppCat‘𝐶)
yoneda.s	⊢ 𝑆 = (SetCat‘𝑈)
yoneda.t	⊢ 𝑇 = (SetCat‘𝑉)
yoneda.q	⊢ 𝑄 = (𝑂 FuncCat 𝑆)
yoneda.h	⊢ 𝐻 = (HomF‘𝑄)
yoneda.r	⊢ 𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇)
yoneda.e	⊢ 𝐸 = (𝑂 evalF 𝑆)
yoneda.z	⊢ 𝑍 = (𝐻 ∘func ((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))
yoneda.c	⊢ (𝜑 → 𝐶 ∈ Cat)
yoneda.w	⊢ (𝜑 → 𝑉 ∈ 𝑊)
yoneda.u	⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ 𝑈)
yoneda.v	⊢ (𝜑 → (ran (Homf ‘𝑄) ∪ 𝑈) ⊆ 𝑉)
yonedalem21.f	⊢ (𝜑 → 𝐹 ∈ (𝑂 Func 𝑆))
yonedalem21.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
yonedalem4.n	⊢ 𝑁 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥 ∈ 𝐵 ↦ (𝑢 ∈ ((1st ‘𝑓)‘𝑥) ↦ (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd ‘𝑓)𝑦)‘𝑔)‘𝑢)))))
yonedalem4.p	⊢ (𝜑 → 𝐴 ∈ ((1st ‘𝐹)‘𝑋))

Assertion

Ref	Expression
yonedalem4a	⊢ (𝜑 → ((𝐹𝑁𝑋)‘𝐴) = (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑦)‘𝑔)‘𝐴))))

Distinct variable groups:   𝑓,𝑔,𝑥,𝑦, 1   𝑢,𝑔,𝐴,𝑦   𝑢,𝑓,𝐶,𝑔,𝑥,𝑦   𝑓,𝐸,𝑔,𝑢,𝑦   𝑓,𝐹,𝑔,𝑢,𝑥,𝑦   𝐵,𝑓,𝑔,𝑢,𝑥,𝑦   𝑓,𝑂,𝑔,𝑢,𝑥,𝑦   𝑆,𝑓,𝑔,𝑢,𝑥,𝑦   𝑄,𝑓,𝑔,𝑢,𝑥   𝑇,𝑓,𝑔,𝑢,𝑦   𝜑,𝑓,𝑔,𝑢,𝑥,𝑦   𝑢,𝑅   𝑓,𝑌,𝑔,𝑢,𝑥,𝑦   𝑓,𝑍,𝑔,𝑢,𝑥,𝑦   𝑓,𝑋,𝑔,𝑢,𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑓)   𝑄(𝑦)   𝑅(𝑥,𝑦,𝑓,𝑔)   𝑇(𝑥)   𝑈(𝑥,𝑦,𝑢,𝑓,𝑔)   1 (𝑢)   𝐸(𝑥)   𝐻(𝑥,𝑦,𝑢,𝑓,𝑔)   𝑁(𝑥,𝑦,𝑢,𝑓,𝑔)   𝑉(𝑥,𝑦,𝑢,𝑓,𝑔)   𝑊(𝑥,𝑦,𝑢,𝑓,𝑔)

Proof of Theorem yonedalem4a

Step	Hyp	Ref	Expression
1		yonedalem4.n	. . . 4 ⊢ 𝑁 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥 ∈ 𝐵 ↦ (𝑢 ∈ ((1st ‘𝑓)‘𝑥) ↦ (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd ‘𝑓)𝑦)‘𝑔)‘𝑢)))))
2	1	a1i 11	. . 3 ⊢ (𝜑 → 𝑁 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥 ∈ 𝐵 ↦ (𝑢 ∈ ((1st ‘𝑓)‘𝑥) ↦ (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd ‘𝑓)𝑦)‘𝑔)‘𝑢))))))
3		simprl 771	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → 𝑓 = 𝐹)
4	3	fveq2d 6833	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → (1st ‘𝑓) = (1st ‘𝐹))
5		simprr 773	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → 𝑥 = 𝑋)
6	4, 5	fveq12d 6836	. . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → ((1st ‘𝑓)‘𝑥) = ((1st ‘𝐹)‘𝑋))
7		simplrr 778	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) ∧ 𝑦 ∈ 𝐵) → 𝑥 = 𝑋)
8	7	oveq2d 7372	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) ∧ 𝑦 ∈ 𝐵) → (𝑦(Hom ‘𝐶)𝑥) = (𝑦(Hom ‘𝐶)𝑋))
9		simplrl 777	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) ∧ 𝑦 ∈ 𝐵) → 𝑓 = 𝐹)
10	9	fveq2d 6833	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) ∧ 𝑦 ∈ 𝐵) → (2nd ‘𝑓) = (2nd ‘𝐹))
11		eqidd 2736	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) ∧ 𝑦 ∈ 𝐵) → 𝑦 = 𝑦)
12	10, 7, 11	oveq123d 7377	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) ∧ 𝑦 ∈ 𝐵) → (𝑥(2nd ‘𝑓)𝑦) = (𝑋(2nd ‘𝐹)𝑦))
13	12	fveq1d 6831	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) ∧ 𝑦 ∈ 𝐵) → ((𝑥(2nd ‘𝑓)𝑦)‘𝑔) = ((𝑋(2nd ‘𝐹)𝑦)‘𝑔))
14	13	fveq1d 6831	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) ∧ 𝑦 ∈ 𝐵) → (((𝑥(2nd ‘𝑓)𝑦)‘𝑔)‘𝑢) = (((𝑋(2nd ‘𝐹)𝑦)‘𝑔)‘𝑢))
15	8, 14	mpteq12dv 5161	. . . . 5 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) ∧ 𝑦 ∈ 𝐵) → (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd ‘𝑓)𝑦)‘𝑔)‘𝑢)) = (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑦)‘𝑔)‘𝑢)))
16	15	mpteq2dva 5167	. . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd ‘𝑓)𝑦)‘𝑔)‘𝑢))) = (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑦)‘𝑔)‘𝑢))))
17	6, 16	mpteq12dv 5161	. . 3 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → (𝑢 ∈ ((1st ‘𝑓)‘𝑥) ↦ (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd ‘𝑓)𝑦)‘𝑔)‘𝑢)))) = (𝑢 ∈ ((1st ‘𝐹)‘𝑋) ↦ (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑦)‘𝑔)‘𝑢)))))
18		yonedalem21.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑂 Func 𝑆))
19		yonedalem21.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
20		fvex 6842	. . . . 5 ⊢ ((1st ‘𝐹)‘𝑋) ∈ V
21	20	mptex 7167	. . . 4 ⊢ (𝑢 ∈ ((1st ‘𝐹)‘𝑋) ↦ (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑦)‘𝑔)‘𝑢)))) ∈ V
22	21	a1i 11	. . 3 ⊢ (𝜑 → (𝑢 ∈ ((1st ‘𝐹)‘𝑋) ↦ (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑦)‘𝑔)‘𝑢)))) ∈ V)
23	2, 17, 18, 19, 22	ovmpod 7508	. 2 ⊢ (𝜑 → (𝐹𝑁𝑋) = (𝑢 ∈ ((1st ‘𝐹)‘𝑋) ↦ (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑦)‘𝑔)‘𝑢)))))
24		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 = 𝐴) → 𝑢 = 𝐴)
25	24	fveq2d 6833	. . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝐴) → (((𝑋(2nd ‘𝐹)𝑦)‘𝑔)‘𝑢) = (((𝑋(2nd ‘𝐹)𝑦)‘𝑔)‘𝐴))
26	25	mpteq2dv 5168	. . 3 ⊢ ((𝜑 ∧ 𝑢 = 𝐴) → (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑦)‘𝑔)‘𝑢)) = (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑦)‘𝑔)‘𝐴)))
27	26	mpteq2dv 5168	. 2 ⊢ ((𝜑 ∧ 𝑢 = 𝐴) → (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑦)‘𝑔)‘𝑢))) = (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑦)‘𝑔)‘𝐴))))
28		yonedalem4.p	. 2 ⊢ (𝜑 → 𝐴 ∈ ((1st ‘𝐹)‘𝑋))
29		yoneda.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
30	29	fvexi 6843	. . . 4 ⊢ 𝐵 ∈ V
31	30	mptex 7167	. . 3 ⊢ (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑦)‘𝑔)‘𝐴))) ∈ V
32	31	a1i 11	. 2 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑦)‘𝑔)‘𝐴))) ∈ V)
33	23, 27, 28, 32	fvmptd 6944	1 ⊢ (𝜑 → ((𝐹𝑁𝑋)‘𝐴) = (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑦)‘𝑔)‘𝐴))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3427   ∪ cun 3883   ⊆ wss 3885  ⟨cop 4563   ↦ cmpt 5155  ran crn 5621  ‘cfv 6487  (class class class)co 7356   ∈ cmpo 7358  1^st c1st 7929  2^nd c2nd 7930  tpos ctpos 8164  Basecbs 17168  Hom chom 17220  Catccat 17619  Idccid 17620  Hom_f chomf 17621  oppCatcoppc 17666   Func cfunc 17810   ∘_func ccofu 17812   FuncCat cfuc 17901  SetCatcsetc 18031   ×_c cxpc 18123   1^st_F c1stf 18124   2^nd_F c2ndf 18125   ⟨,⟩_F cprf 18126   eval_F cevlf 18164  Hom_Fchof 18203  Yoncyon 18204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pr 5364

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-ral 3050  df-rex 3060  df-reu 3341  df-rab 3388  df-v 3429  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-ov 7359  df-oprab 7360  df-mpo 7361

This theorem is referenced by:  yonedalem4b  18231  yonedalem4c  18232  yonffthlem  18237