Theorem curf12 18176

Description: The partially evaluated curry functor at a morphism. (Contributed by Mario Carneiro, 12-Jan-2017.)

Hypotheses

Ref	Expression
curfval.g	⊢ 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)
curfval.a	⊢ 𝐴 = (Base‘𝐶)
curfval.c	⊢ (𝜑 → 𝐶 ∈ Cat)
curfval.d	⊢ (𝜑 → 𝐷 ∈ Cat)
curfval.f	⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
curfval.b	⊢ 𝐵 = (Base‘𝐷)
curf1.x	⊢ (𝜑 → 𝑋 ∈ 𝐴)
curf1.k	⊢ 𝐾 = ((1st ‘𝐺)‘𝑋)
curf11.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
curf12.j	⊢ 𝐽 = (Hom ‘𝐷)
curf12.1	⊢ 1 = (Id‘𝐶)
curf12.y	⊢ (𝜑 → 𝑍 ∈ 𝐵)
curf12.g	⊢ (𝜑 → 𝐻 ∈ (𝑌𝐽𝑍))

Assertion

Ref	Expression
curf12	⊢ (𝜑 → ((𝑌(2nd ‘𝐾)𝑍)‘𝐻) = (( 1 ‘𝑋)(⟨𝑋, 𝑌⟩(2nd ‘𝐹)⟨𝑋, 𝑍⟩)𝐻))

Proof of Theorem curf12

Dummy variables 𝑔 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		curfval.g	. . . 4 ⊢ 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)
2		curfval.a	. . . 4 ⊢ 𝐴 = (Base‘𝐶)
3		curfval.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
4		curfval.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
5		curfval.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
6		curfval.b	. . . 4 ⊢ 𝐵 = (Base‘𝐷)
7		curf1.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
8		curf1.k	. . . 4 ⊢ 𝐾 = ((1st ‘𝐺)‘𝑋)
9		curf12.j	. . . 4 ⊢ 𝐽 = (Hom ‘𝐷)
10		curf12.1	. . . 4 ⊢ 1 = (Id‘𝐶)
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	curf1 18174	. . 3 ⊢ (𝜑 → 𝐾 = ⟨(𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔)))⟩)
12	6	fvexi 6902	. . . . 5 ⊢ 𝐵 ∈ V
13	12	mptex 7221	. . . 4 ⊢ (𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘𝐹)𝑦)) ∈ V
14	12, 12	mpoex 8062	. . . 4 ⊢ (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔))) ∈ V
15	13, 14	op2ndd 7982	. . 3 ⊢ (𝐾 = ⟨(𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔)))⟩ → (2nd ‘𝐾) = (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔))))
16	11, 15	syl 17	. 2 ⊢ (𝜑 → (2nd ‘𝐾) = (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔))))
17		curf11.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
18		curf12.y	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
19	18	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑦 = 𝑌) → 𝑍 ∈ 𝐵)
20		ovex 7438	. . . . 5 ⊢ (𝑦𝐽𝑧) ∈ V
21	20	mptex 7221	. . . 4 ⊢ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔)) ∈ V
22	21	a1i 11	. . 3 ⊢ ((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) → (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔)) ∈ V)
23		curf12.g	. . . . . 6 ⊢ (𝜑 → 𝐻 ∈ (𝑌𝐽𝑍))
24	23	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) → 𝐻 ∈ (𝑌𝐽𝑍))
25		simprl 769	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) → 𝑦 = 𝑌)
26		simprr 771	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) → 𝑧 = 𝑍)
27	25, 26	oveq12d 7423	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) → (𝑦𝐽𝑧) = (𝑌𝐽𝑍))
28	24, 27	eleqtrrd 2836	. . . 4 ⊢ ((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) → 𝐻 ∈ (𝑦𝐽𝑧))
29		ovexd 7440	. . . 4 ⊢ (((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔) ∈ V)
30		simplrl 775	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → 𝑦 = 𝑌)
31	30	opeq2d 4879	. . . . . 6 ⊢ (((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → ⟨𝑋, 𝑦⟩ = ⟨𝑋, 𝑌⟩)
32		simplrr 776	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → 𝑧 = 𝑍)
33	32	opeq2d 4879	. . . . . 6 ⊢ (((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → ⟨𝑋, 𝑧⟩ = ⟨𝑋, 𝑍⟩)
34	31, 33	oveq12d 7423	. . . . 5 ⊢ (((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → (⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩) = (⟨𝑋, 𝑌⟩(2nd ‘𝐹)⟨𝑋, 𝑍⟩))
35		eqidd 2733	. . . . 5 ⊢ (((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → ( 1 ‘𝑋) = ( 1 ‘𝑋))
36		simpr 485	. . . . 5 ⊢ (((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → 𝑔 = 𝐻)
37	34, 35, 36	oveq123d 7426	. . . 4 ⊢ (((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔) = (( 1 ‘𝑋)(⟨𝑋, 𝑌⟩(2nd ‘𝐹)⟨𝑋, 𝑍⟩)𝐻))
38	28, 29, 37	fvmptdv2 7013	. . 3 ⊢ ((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) → ((𝑌(2nd ‘𝐾)𝑍) = (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔)) → ((𝑌(2nd ‘𝐾)𝑍)‘𝐻) = (( 1 ‘𝑋)(⟨𝑋, 𝑌⟩(2nd ‘𝐹)⟨𝑋, 𝑍⟩)𝐻)))
39	17, 19, 22, 38	ovmpodv 7561	. 2 ⊢ (𝜑 → ((2nd ‘𝐾) = (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔))) → ((𝑌(2nd ‘𝐾)𝑍)‘𝐻) = (( 1 ‘𝑋)(⟨𝑋, 𝑌⟩(2nd ‘𝐹)⟨𝑋, 𝑍⟩)𝐻)))
40	16, 39	mpd 15	1 ⊢ (𝜑 → ((𝑌(2nd ‘𝐾)𝑍)‘𝐻) = (( 1 ‘𝑋)(⟨𝑋, 𝑌⟩(2nd ‘𝐹)⟨𝑋, 𝑍⟩)𝐻))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3474  ⟨cop 4633   ↦ cmpt 5230  ‘cfv 6540  (class class class)co 7405   ∈ cmpo 7407  1^st c1st 7969  2^nd c2nd 7970  Basecbs 17140  Hom chom 17204  Catccat 17604  Idccid 17605   Func cfunc 17800   ×_c cxpc 18116   curry_F ccurf 18159

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972  df-curf 18163

This theorem is referenced by:  curf1cl  18177  curf2cl  18180  uncfcurf  18188  diag12  18193  yon12  18214