Theorem uncov 37773

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 5100	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩uncurry 𝐹𝑤 ↔ ⟨⟨𝐴, 𝐵⟩, 𝑤⟩ ∈ uncurry 𝐹)
2		df-unc 8212	. . . . . 6 ⊢ uncurry 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹‘𝑥)𝑧}
3	2	eleq2i 2829	. . . . 5 ⊢ (⟨⟨𝐴, 𝐵⟩, 𝑤⟩ ∈ uncurry 𝐹 ↔ ⟨⟨𝐴, 𝐵⟩, 𝑤⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹‘𝑥)𝑧})
4	1, 3	bitri 275	. . . 4 ⊢ (⟨𝐴, 𝐵⟩uncurry 𝐹𝑤 ↔ ⟨⟨𝐴, 𝐵⟩, 𝑤⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹‘𝑥)𝑧})
5		vex 3445	. . . . 5 ⊢ 𝑤 ∈ V
6		simp2 1138	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝑤) → 𝑦 = 𝐵)
7		fveq2 6835	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
8	7	3ad2ant1 1134	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝑤) → (𝐹‘𝑥) = (𝐹‘𝐴))
9		simp3 1139	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝑤) → 𝑧 = 𝑤)
10	6, 8, 9	breq123d 5113	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝑤) → (𝑦(𝐹‘𝑥)𝑧 ↔ 𝐵(𝐹‘𝐴)𝑤))
11	10	eloprabga 7469	. . . . 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 1453	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨⟨𝐴, 𝐵⟩, 𝑤⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹‘𝑥)𝑧} ↔ 𝐵(𝐹‘𝐴)𝑤))
13	4, 12	bitrid 283	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩uncurry 𝐹𝑤 ↔ 𝐵(𝐹‘𝐴)𝑤))
14	13	iotabidv 6477	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (℩𝑤⟨𝐴, 𝐵⟩uncurry 𝐹𝑤) = (℩𝑤𝐵(𝐹‘𝐴)𝑤))
15		df-ov 7363	. . 3 ⊢ (𝐴uncurry 𝐹𝐵) = (uncurry 𝐹‘⟨𝐴, 𝐵⟩)
16		df-fv 6501	. . 3 ⊢ (uncurry 𝐹‘⟨𝐴, 𝐵⟩) = (℩𝑤⟨𝐴, 𝐵⟩uncurry 𝐹𝑤)
17	15, 16	eqtri 2760	. 2 ⊢ (𝐴uncurry 𝐹𝐵) = (℩𝑤⟨𝐴, 𝐵⟩uncurry 𝐹𝑤)
18		df-fv 6501	. 2 ⊢ ((𝐹‘𝐴)‘𝐵) = (℩𝑤𝐵(𝐹‘𝐴)𝑤)
19	14, 17, 18	3eqtr4g 2797	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴uncurry 𝐹𝐵) = ((𝐹‘𝐴)‘𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  Vcvv 3441  ⟨cop 4587   class class class wbr 5099  ℩cio 6447  ‘cfv 6493  (class class class)co 7360  {coprab 7361  uncurry cunc 8210

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-iota 6449  df-fv 6501  df-ov 7363  df-oprab 7364  df-unc 8212

This theorem is referenced by:  curunc  37774  matunitlindflem2  37789