Theorem yonedalem4a 18337

Description: Lemma for yoneda 18345. (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 782	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → 𝑓 = 𝐹)
4	3	fveq2d 6885	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → (1st ‘𝑓) = (1st ‘𝐹))
5		simprr 784	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → 𝑥 = 𝑋)
6	4, 5	fveq12d 6888	. . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → ((1st ‘𝑓)‘𝑥) = ((1st ‘𝐹)‘𝑋))
7		simplrr 789	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) ∧ 𝑦 ∈ 𝐵) → 𝑥 = 𝑋)
8	7	oveq2d 7428	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) ∧ 𝑦 ∈ 𝐵) → (𝑦(Hom ‘𝐶)𝑥) = (𝑦(Hom ‘𝐶)𝑋))
9		simplrl 788	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) ∧ 𝑦 ∈ 𝐵) → 𝑓 = 𝐹)
10	9	fveq2d 6885	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) ∧ 𝑦 ∈ 𝐵) → (2nd ‘𝑓) = (2nd ‘𝐹))
11		eqidd 2763	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) ∧ 𝑦 ∈ 𝐵) → 𝑦 = 𝑦)
12	10, 7, 11	oveq123d 7433	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) ∧ 𝑦 ∈ 𝐵) → (𝑥(2nd ‘𝑓)𝑦) = (𝑋(2nd ‘𝐹)𝑦))
13	12	fveq1d 6883	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) ∧ 𝑦 ∈ 𝐵) → ((𝑥(2nd ‘𝑓)𝑦)‘𝑔) = ((𝑋(2nd ‘𝐹)𝑦)‘𝑔))
14	13	fveq1d 6883	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) ∧ 𝑦 ∈ 𝐵) → (((𝑥(2nd ‘𝑓)𝑦)‘𝑔)‘𝑢) = (((𝑋(2nd ‘𝐹)𝑦)‘𝑔)‘𝑢))
15	8, 14	mpteq12dv 5197	. . . . 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 5203	. . . 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 5197	. . 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 6894	. . . . 5 ⊢ ((1st ‘𝐹)‘𝑋) ∈ V
21	20	mptex 7221	. . . 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 7564	. 2 ⊢ (𝜑 → (𝐹𝑁𝑋) = (𝑢 ∈ ((1st ‘𝐹)‘𝑋) ↦ (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑦)‘𝑔)‘𝑢)))))
24		simpr 489	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 = 𝐴) → 𝑢 = 𝐴)
25	24	fveq2d 6885	. . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝐴) → (((𝑋(2nd ‘𝐹)𝑦)‘𝑔)‘𝑢) = (((𝑋(2nd ‘𝐹)𝑦)‘𝑔)‘𝐴))
26	25	mpteq2dv 5204	. . 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 5204	. 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 6895	. . . 4 ⊢ 𝐵 ∈ V
31	30	mptex 7221	. . 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 6997	1 ⊢ (𝜑 → ((𝐹𝑁𝑋)‘𝐴) = (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑦)‘𝑔)‘𝐴))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1569   ∈ wcel 2142  Vcvv 3454   ∪ cun 3902   ⊆ wss 3904  ⟨cop 4594   ↦ cmpt 5191  ran crn 5661  ‘cfv 6536  (class class class)co 7412   ∈ cmpo 7414  1^st c1st 7982  2^nd c2nd 7983  tpos ctpos 8219  Basecbs 17275  Hom chom 17327  Catccat 17726  Idccid 17727  Hom_f chomf 17728  oppCatcoppc 17773   Func cfunc 17917   ∘_func ccofu 17919   FuncCat cfuc 18008  SetCatcsetc 18138   ×_c cxpc 18230   1^st_F c1stf 18231   2^nd_F c2ndf 18232   ⟨,⟩_F cprf 18233   eval_F cevlf 18271  Hom_Fchof 18310  Yoncyon 18311

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7415  df-oprab 7416  df-mpo 7417

This theorem is used by:  yonedalem4b  18338  yonedalem4c  18339  yonffthlem  18344