Theorem curf1 18131

Description: Value of the object part of the curry functor. (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)
curf1.j	⊢ 𝐽 = (Hom ‘𝐷)
curf1.1	⊢ 1 = (Id‘𝐶)

Assertion

Ref	Expression
curf1	⊢ (𝜑 → 𝐾 = ⟨(𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔)))⟩)

Distinct variable groups:   𝑦,𝑔,𝑧, 1   𝑦,𝐴   𝐵,𝑔,𝑦,𝑧   𝐶,𝑔,𝑦,𝑧   𝐷,𝑔,𝑦,𝑧   𝜑,𝑔,𝑦,𝑧   𝑔,𝐸,𝑦,𝑧   𝑔,𝐽   𝑔,𝐾,𝑦,𝑧   𝑔,𝑋,𝑦,𝑧   𝑔,𝐹,𝑦,𝑧

Allowed substitution hints:   𝐴(𝑧,𝑔)   𝐺(𝑦,𝑧,𝑔)   𝐽(𝑦,𝑧)

Proof of Theorem curf1

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		curf1.k	. 2 ⊢ 𝐾 = ((1st ‘𝐺)‘𝑋)
2		curfval.g	. . . 4 ⊢ 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)
3		curfval.a	. . . 4 ⊢ 𝐴 = (Base‘𝐶)
4		curfval.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
5		curfval.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
6		curfval.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
7		curfval.b	. . . 4 ⊢ 𝐵 = (Base‘𝐷)
8		curf1.j	. . . 4 ⊢ 𝐽 = (Hom ‘𝐷)
9		curf1.1	. . . 4 ⊢ 1 = (Id‘𝐶)
10	2, 3, 4, 5, 6, 7, 8, 9	curf1fval 18130	. . 3 ⊢ (𝜑 → (1st ‘𝐺) = (𝑥 ∈ 𝐴 ↦ ⟨(𝑦 ∈ 𝐵 ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩))
11		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
12	11	oveq1d 7361	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑥(1st ‘𝐹)𝑦) = (𝑋(1st ‘𝐹)𝑦))
13	12	mpteq2dv 5185	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑦 ∈ 𝐵 ↦ (𝑥(1st ‘𝐹)𝑦)) = (𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘𝐹)𝑦)))
14		simp1r 1199	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → 𝑥 = 𝑋)
15	14	opeq1d 4831	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑦⟩)
16	14	opeq1d 4831	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ⟨𝑥, 𝑧⟩ = ⟨𝑋, 𝑧⟩)
17	15, 16	oveq12d 7364	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩) = (⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩))
18	14	fveq2d 6826	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ( 1 ‘𝑥) = ( 1 ‘𝑋))
19		eqidd 2732	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → 𝑔 = 𝑔)
20	17, 18, 19	oveq123d 7367	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (( 1 ‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔) = (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔))
21	20	mpteq2dv 5185	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)) = (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔)))
22	21	mpoeq3dva 7423	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔))) = (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔))))
23	13, 22	opeq12d 4833	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ⟨(𝑦 ∈ 𝐵 ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩ = ⟨(𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔)))⟩)
24		curf1.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
25		opex 5404	. . . 4 ⊢ ⟨(𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔)))⟩ ∈ V
26	25	a1i 11	. . 3 ⊢ (𝜑 → ⟨(𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔)))⟩ ∈ V)
27	10, 23, 24, 26	fvmptd 6936	. 2 ⊢ (𝜑 → ((1st ‘𝐺)‘𝑋) = ⟨(𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔)))⟩)
28	1, 27	eqtrid 2778	1 ⊢ (𝜑 → 𝐾 = ⟨(𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔)))⟩)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  Vcvv 3436  ⟨cop 4582   ↦ cmpt 5172  ‘cfv 6481  (class class class)co 7346   ∈ cmpo 7348  1^st c1st 7919  2^nd c2nd 7920  Basecbs 17120  Hom chom 17172  Catccat 17570  Idccid 17571   Func cfunc 17761   ×_c cxpc 18074   curry_F ccurf 18116

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 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

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 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  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  df-1st 7921  df-2nd 7922  df-curf 18120

This theorem is referenced by:  curf11  18132  curf12  18133  curf1cl  18134  curf2ndf  18153  tposcurf1  49337  postcofval  49402