Theorem curf12 17861

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 17859	. . 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 6770	. . . . 5 ⊢ 𝐵 ∈ V
13	12	mptex 7081	. . . 4 ⊢ (𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘𝐹)𝑦)) ∈ V
14	12, 12	mpoex 7893	. . . 4 ⊢ (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔))) ∈ V
15	13, 14	op2ndd 7815	. . 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 480	. . 3 ⊢ ((𝜑 ∧ 𝑦 = 𝑌) → 𝑍 ∈ 𝐵)
20		ovex 7288	. . . . 5 ⊢ (𝑦𝐽𝑧) ∈ V
21	20	mptex 7081	. . . 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 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) → 𝐻 ∈ (𝑌𝐽𝑍))
25		simprl 767	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) → 𝑦 = 𝑌)
26		simprr 769	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) → 𝑧 = 𝑍)
27	25, 26	oveq12d 7273	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) → (𝑦𝐽𝑧) = (𝑌𝐽𝑍))
28	24, 27	eleqtrrd 2842	. . . 4 ⊢ ((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) → 𝐻 ∈ (𝑦𝐽𝑧))
29		ovexd 7290	. . . 4 ⊢ (((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔) ∈ V)
30		simplrl 773	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → 𝑦 = 𝑌)
31	30	opeq2d 4808	. . . . . 6 ⊢ (((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → ⟨𝑋, 𝑦⟩ = ⟨𝑋, 𝑌⟩)
32		simplrr 774	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → 𝑧 = 𝑍)
33	32	opeq2d 4808	. . . . . 6 ⊢ (((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → ⟨𝑋, 𝑧⟩ = ⟨𝑋, 𝑍⟩)
34	31, 33	oveq12d 7273	. . . . 5 ⊢ (((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → (⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩) = (⟨𝑋, 𝑌⟩(2nd ‘𝐹)⟨𝑋, 𝑍⟩))
35		eqidd 2739	. . . . 5 ⊢ (((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → ( 1 ‘𝑋) = ( 1 ‘𝑋))
36		simpr 484	. . . . 5 ⊢ (((𝜑 ∧ (𝑦 = 𝑌 ∧ 𝑧 = 𝑍)) ∧ 𝑔 = 𝐻) → 𝑔 = 𝐻)
37	34, 35, 36	oveq123d 7276	. . . 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 6875	. . 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 7408	. 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 395   = wceq 1539   ∈ wcel 2108  Vcvv 3422  ⟨cop 4564   ↦ cmpt 5153  ‘cfv 6418  (class class class)co 7255   ∈ cmpo 7257  1^st c1st 7802  2^nd c2nd 7803  Basecbs 16840  Hom chom 16899  Catccat 17290  Idccid 17291   Func cfunc 17485   ×_c cxpc 17801   curry_F ccurf 17844

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-curf 17848

This theorem is referenced by:  curf1cl  17862  curf2cl  17865  uncfcurf  17873  diag12  17878  yon12  17899