Theorem uncov 8921

Description: Value of uncurrying. (Contributed by Brendan Leahy, 2-Jun-2021.)

Assertion

Ref	Expression
uncov	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴uncurry 𝐹𝐵) = ((𝐹‘𝐴)‘𝐵))

Proof of Theorem uncov

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

Step	Hyp	Ref	Expression
1		df-br 5107	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩uncurry 𝐹𝑤 ↔ ⟨⟨𝐴, 𝐵⟩, 𝑤⟩ ∈ uncurry 𝐹)
2		df-unc 8313	. . . . . 6 ⊢ uncurry 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹‘𝑥)𝑧}
3	2	eleq2i 2856	. . . . 5 ⊢ (⟨⟨𝐴, 𝐵⟩, 𝑤⟩ ∈ uncurry 𝐹 ↔ ⟨⟨𝐴, 𝐵⟩, 𝑤⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹‘𝑥)𝑧})
4	1, 3	bitri 278	. . . 4 ⊢ (⟨𝐴, 𝐵⟩uncurry 𝐹𝑤 ↔ ⟨⟨𝐴, 𝐵⟩, 𝑤⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹‘𝑥)𝑧})
5		vex 3458	. . . . 5 ⊢ 𝑤 ∈ V
6		simp2 1155	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝑤) → 𝑦 = 𝐵)
7		fveq2 6908	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
8	7	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝑤) → (𝐹‘𝑥) = (𝐹‘𝐴))
9		simp3 1156	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝑤) → 𝑧 = 𝑤)
10	6, 8, 9	breq123d 5120	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝑤) → (𝑦(𝐹‘𝑥)𝑧 ↔ 𝐵(𝐹‘𝐴)𝑤))
11	10	eloprabga 7552	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑤 ∈ V) → (⟨⟨𝐴, 𝐵⟩, 𝑤⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹‘𝑥)𝑧} ↔ 𝐵(𝐹‘𝐴)𝑤))
12	5, 11	mp3an3 1480	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨⟨𝐴, 𝐵⟩, 𝑤⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹‘𝑥)𝑧} ↔ 𝐵(𝐹‘𝐴)𝑤))
13	4, 12	bitrid 286	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩uncurry 𝐹𝑤 ↔ 𝐵(𝐹‘𝐴)𝑤))
14	13	iotabidv 6541	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (℩𝑤⟨𝐴, 𝐵⟩uncurry 𝐹𝑤) = (℩𝑤𝐵(𝐹‘𝐴)𝑤))
15		df-ov 7446	. . 3 ⊢ (𝐴uncurry 𝐹𝐵) = (uncurry 𝐹‘⟨𝐴, 𝐵⟩)
16		df-fv 6566	. . 3 ⊢ (uncurry 𝐹‘⟨𝐴, 𝐵⟩) = (℩𝑤⟨𝐴, 𝐵⟩uncurry 𝐹𝑤)
17	15, 16	eqtri 2786	. 2 ⊢ (𝐴uncurry 𝐹𝐵) = (℩𝑤⟨𝐴, 𝐵⟩uncurry 𝐹𝑤)
18		df-fv 6566	. 2 ⊢ ((𝐹‘𝐴)‘𝐵) = (℩𝑤𝐵(𝐹‘𝐴)𝑤)
19	14, 17, 18	3eqtr4g 2823	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴uncurry 𝐹𝐵) = ((𝐹‘𝐴)‘𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  Vcvv 3454  ⟨cop 4838   class class class wbr 5106  ℩cio 6510  ‘cfv 6558  (class class class)co 7443  {coprab 7444  uncurry cunc 8311

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-iota 6512  df-fv 6566  df-ov 7446  df-oprab 7447  df-unc 8313

This theorem is used by:  matunitlindflem2  23025  curunc  38717