Theorem yonedalem4b 18182

Description: Lemma for yoneda 18189. (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))
yonedalem4b.p	⊢ (𝜑 → 𝑃 ∈ 𝐵)
yonedalem4b.g	⊢ (𝜑 → 𝐺 ∈ (𝑃(Hom ‘𝐶)𝑋))

Assertion

Ref	Expression
yonedalem4b	⊢ (𝜑 → ((((𝐹𝑁𝑋)‘𝐴)‘𝑃)‘𝐺) = (((𝑋(2nd ‘𝐹)𝑃)‘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,𝐺,𝑔,𝑥,𝑦   𝑓,𝑂,𝑔,𝑢,𝑥,𝑦   𝑆,𝑓,𝑔,𝑢,𝑥,𝑦   𝑄,𝑓,𝑔,𝑢,𝑥   𝑇,𝑓,𝑔,𝑢,𝑦   𝑃,𝑓,𝑔,𝑥,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)   𝑃(𝑢)   𝑄(𝑦)   𝑅(𝑥,𝑦,𝑓,𝑔)   𝑇(𝑥)   𝑈(𝑥,𝑦,𝑢,𝑓,𝑔)   1 (𝑢)   𝐸(𝑥)   𝐺(𝑢)   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 yonedalem4b

Step	Hyp	Ref	Expression
1		yoneda.y	. . . . 5 ⊢ 𝑌 = (Yon‘𝐶)
2		yoneda.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
3		yoneda.1	. . . . 5 ⊢ 1 = (Id‘𝐶)
4		yoneda.o	. . . . 5 ⊢ 𝑂 = (oppCat‘𝐶)
5		yoneda.s	. . . . 5 ⊢ 𝑆 = (SetCat‘𝑈)
6		yoneda.t	. . . . 5 ⊢ 𝑇 = (SetCat‘𝑉)
7		yoneda.q	. . . . 5 ⊢ 𝑄 = (𝑂 FuncCat 𝑆)
8		yoneda.h	. . . . 5 ⊢ 𝐻 = (HomF‘𝑄)
9		yoneda.r	. . . . 5 ⊢ 𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇)
10		yoneda.e	. . . . 5 ⊢ 𝐸 = (𝑂 evalF 𝑆)
11		yoneda.z	. . . . 5 ⊢ 𝑍 = (𝐻 ∘func ((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))
12		yoneda.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
13		yoneda.w	. . . . 5 ⊢ (𝜑 → 𝑉 ∈ 𝑊)
14		yoneda.u	. . . . 5 ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ 𝑈)
15		yoneda.v	. . . . 5 ⊢ (𝜑 → (ran (Homf ‘𝑄) ∪ 𝑈) ⊆ 𝑉)
16		yonedalem21.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝑂 Func 𝑆))
17		yonedalem21.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
18		yonedalem4.n	. . . . 5 ⊢ 𝑁 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥 ∈ 𝐵 ↦ (𝑢 ∈ ((1st ‘𝑓)‘𝑥) ↦ (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd ‘𝑓)𝑦)‘𝑔)‘𝑢)))))
19		yonedalem4.p	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ((1st ‘𝐹)‘𝑋))
20	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19	yonedalem4a 18181	. . . 4 ⊢ (𝜑 → ((𝐹𝑁𝑋)‘𝐴) = (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑦)‘𝑔)‘𝐴))))
21	20	fveq1d 6824	. . 3 ⊢ (𝜑 → (((𝐹𝑁𝑋)‘𝐴)‘𝑃) = ((𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑦)‘𝑔)‘𝐴)))‘𝑃))
22	21	fveq1d 6824	. 2 ⊢ (𝜑 → ((((𝐹𝑁𝑋)‘𝐴)‘𝑃)‘𝐺) = (((𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑦)‘𝑔)‘𝐴)))‘𝑃)‘𝐺))
23		eqidd 2732	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑦)‘𝑔)‘𝐴))) = (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑦)‘𝑔)‘𝐴))))
24		yonedalem4b.p	. . . 4 ⊢ (𝜑 → 𝑃 ∈ 𝐵)
25		ovex 7379	. . . . . 6 ⊢ (𝑦(Hom ‘𝐶)𝑋) ∈ V
26	25	mptex 7157	. . . . 5 ⊢ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑦)‘𝑔)‘𝐴)) ∈ V
27	26	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝑃) → (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑦)‘𝑔)‘𝐴)) ∈ V)
28		yonedalem4b.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ (𝑃(Hom ‘𝐶)𝑋))
29	28	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝑃) → 𝐺 ∈ (𝑃(Hom ‘𝐶)𝑋))
30		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 = 𝑃) → 𝑦 = 𝑃)
31	30	oveq1d 7361	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝑃) → (𝑦(Hom ‘𝐶)𝑋) = (𝑃(Hom ‘𝐶)𝑋))
32	29, 31	eleqtrrd 2834	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝑃) → 𝐺 ∈ (𝑦(Hom ‘𝐶)𝑋))
33		fvexd 6837	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 = 𝑃) ∧ 𝑔 = 𝐺) → (((𝑋(2nd ‘𝐹)𝑦)‘𝑔)‘𝐴) ∈ V)
34		simplr 768	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 = 𝑃) ∧ 𝑔 = 𝐺) → 𝑦 = 𝑃)
35	34	oveq2d 7362	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 = 𝑃) ∧ 𝑔 = 𝐺) → (𝑋(2nd ‘𝐹)𝑦) = (𝑋(2nd ‘𝐹)𝑃))
36		simpr 484	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 = 𝑃) ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
37	35, 36	fveq12d 6829	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 = 𝑃) ∧ 𝑔 = 𝐺) → ((𝑋(2nd ‘𝐹)𝑦)‘𝑔) = ((𝑋(2nd ‘𝐹)𝑃)‘𝐺))
38	37	fveq1d 6824	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 = 𝑃) ∧ 𝑔 = 𝐺) → (((𝑋(2nd ‘𝐹)𝑦)‘𝑔)‘𝐴) = (((𝑋(2nd ‘𝐹)𝑃)‘𝐺)‘𝐴))
39	32, 33, 38	fvmptdv2 6947	. . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝑃) → (((𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑦)‘𝑔)‘𝐴)))‘𝑃) = (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑦)‘𝑔)‘𝐴)) → (((𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑦)‘𝑔)‘𝐴)))‘𝑃)‘𝐺) = (((𝑋(2nd ‘𝐹)𝑃)‘𝐺)‘𝐴)))
40		nfmpt1 5188	. . . 4 ⊢ Ⅎ𝑦(𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑦)‘𝑔)‘𝐴)))
41		nffvmpt1 6833	. . . . . 6 ⊢ Ⅎ𝑦((𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑦)‘𝑔)‘𝐴)))‘𝑃)
42		nfcv 2894	. . . . . 6 ⊢ Ⅎ𝑦𝐺
43	41, 42	nffv 6832	. . . . 5 ⊢ Ⅎ𝑦(((𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑦)‘𝑔)‘𝐴)))‘𝑃)‘𝐺)
44	43	nfeq1 2910	. . . 4 ⊢ Ⅎ𝑦(((𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑦)‘𝑔)‘𝐴)))‘𝑃)‘𝐺) = (((𝑋(2nd ‘𝐹)𝑃)‘𝐺)‘𝐴)
45	24, 27, 39, 40, 44	fvmptd2f 6945	. . 3 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑦)‘𝑔)‘𝐴))) = (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑦)‘𝑔)‘𝐴))) → (((𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑦)‘𝑔)‘𝐴)))‘𝑃)‘𝐺) = (((𝑋(2nd ‘𝐹)𝑃)‘𝐺)‘𝐴)))
46	23, 45	mpd 15	. 2 ⊢ (𝜑 → (((𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑦)‘𝑔)‘𝐴)))‘𝑃)‘𝐺) = (((𝑋(2nd ‘𝐹)𝑃)‘𝐺)‘𝐴))
47	22, 46	eqtrd 2766	1 ⊢ (𝜑 → ((((𝐹𝑁𝑋)‘𝐴)‘𝑃)‘𝐺) = (((𝑋(2nd ‘𝐹)𝑃)‘𝐺)‘𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  Vcvv 3436   ∪ cun 3895   ⊆ wss 3897  ⟨cop 4579   ↦ cmpt 5170  ran crn 5615  ‘cfv 6481  (class class class)co 7346   ∈ cmpo 7348  1^st c1st 7919  2^nd c2nd 7920  tpos ctpos 8155  Basecbs 17120  Hom chom 17172  Catccat 17570  Idccid 17571  Hom_f chomf 17572  oppCatcoppc 17617   Func cfunc 17761   ∘_func ccofu 17763   FuncCat cfuc 17852  SetCatcsetc 17982   ×_c cxpc 18074   1^st_F c1stf 18075   2^nd_F c2ndf 18076   ⟨,⟩_F cprf 18077   eval_F cevlf 18115  Hom_Fchof 18154  Yoncyon 18155

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351

This theorem is referenced by:  yonedalem4c  18183  yonedainv  18187