Theorem uncov 35685

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 5071	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩uncurry 𝐹𝑤 ↔ ⟨⟨𝐴, 𝐵⟩, 𝑤⟩ ∈ uncurry 𝐹)
2		df-unc 8055	. . . . . 6 ⊢ uncurry 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹‘𝑥)𝑧}
3	2	eleq2i 2830	. . . . 5 ⊢ (⟨⟨𝐴, 𝐵⟩, 𝑤⟩ ∈ uncurry 𝐹 ↔ ⟨⟨𝐴, 𝐵⟩, 𝑤⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹‘𝑥)𝑧})
4	1, 3	bitri 274	. . . 4 ⊢ (⟨𝐴, 𝐵⟩uncurry 𝐹𝑤 ↔ ⟨⟨𝐴, 𝐵⟩, 𝑤⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹‘𝑥)𝑧})
5		vex 3426	. . . . 5 ⊢ 𝑤 ∈ V
6		simp2 1135	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝑤) → 𝑦 = 𝐵)
7		fveq2 6756	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
8	7	3ad2ant1 1131	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝑤) → (𝐹‘𝑥) = (𝐹‘𝐴))
9		simp3 1136	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝑤) → 𝑧 = 𝑤)
10	6, 8, 9	breq123d 5084	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝑤) → (𝑦(𝐹‘𝑥)𝑧 ↔ 𝐵(𝐹‘𝐴)𝑤))
11	10	eloprabga 7360	. . . . 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 1448	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨⟨𝐴, 𝐵⟩, 𝑤⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹‘𝑥)𝑧} ↔ 𝐵(𝐹‘𝐴)𝑤))
13	4, 12	syl5bb 282	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩uncurry 𝐹𝑤 ↔ 𝐵(𝐹‘𝐴)𝑤))
14	13	iotabidv 6402	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (℩𝑤⟨𝐴, 𝐵⟩uncurry 𝐹𝑤) = (℩𝑤𝐵(𝐹‘𝐴)𝑤))
15		df-ov 7258	. . 3 ⊢ (𝐴uncurry 𝐹𝐵) = (uncurry 𝐹‘⟨𝐴, 𝐵⟩)
16		df-fv 6426	. . 3 ⊢ (uncurry 𝐹‘⟨𝐴, 𝐵⟩) = (℩𝑤⟨𝐴, 𝐵⟩uncurry 𝐹𝑤)
17	15, 16	eqtri 2766	. 2 ⊢ (𝐴uncurry 𝐹𝐵) = (℩𝑤⟨𝐴, 𝐵⟩uncurry 𝐹𝑤)
18		df-fv 6426	. 2 ⊢ ((𝐹‘𝐴)‘𝐵) = (℩𝑤𝐵(𝐹‘𝐴)𝑤)
19	14, 17, 18	3eqtr4g 2804	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴uncurry 𝐹𝐵) = ((𝐹‘𝐴)‘𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  Vcvv 3422  ⟨cop 4564   class class class wbr 5070  ℩cio 6374  ‘cfv 6418  (class class class)co 7255  {coprab 7256  uncurry cunc 8053

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-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

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-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-iota 6376  df-fv 6426  df-ov 7258  df-oprab 7259  df-unc 8055

This theorem is referenced by:  curunc  35686  matunitlindflem2  35701