Theorem uncov 35754

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 5080	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩uncurry 𝐹𝑤 ↔ ⟨⟨𝐴, 𝐵⟩, 𝑤⟩ ∈ uncurry 𝐹)
2		df-unc 8075	. . . . . 6 ⊢ uncurry 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹‘𝑥)𝑧}
3	2	eleq2i 2832	. . . . 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 3435	. . . . 5 ⊢ 𝑤 ∈ V
6		simp2 1136	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝑤) → 𝑦 = 𝐵)
7		fveq2 6771	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
8	7	3ad2ant1 1132	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝑤) → (𝐹‘𝑥) = (𝐹‘𝐴))
9		simp3 1137	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝑤) → 𝑧 = 𝑤)
10	6, 8, 9	breq123d 5093	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝑤) → (𝑦(𝐹‘𝑥)𝑧 ↔ 𝐵(𝐹‘𝐴)𝑤))
11	10	eloprabga 7376	. . . . 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 1449	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨⟨𝐴, 𝐵⟩, 𝑤⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹‘𝑥)𝑧} ↔ 𝐵(𝐹‘𝐴)𝑤))
13	4, 12	syl5bb 283	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩uncurry 𝐹𝑤 ↔ 𝐵(𝐹‘𝐴)𝑤))
14	13	iotabidv 6416	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (℩𝑤⟨𝐴, 𝐵⟩uncurry 𝐹𝑤) = (℩𝑤𝐵(𝐹‘𝐴)𝑤))
15		df-ov 7274	. . 3 ⊢ (𝐴uncurry 𝐹𝐵) = (uncurry 𝐹‘⟨𝐴, 𝐵⟩)
16		df-fv 6440	. . 3 ⊢ (uncurry 𝐹‘⟨𝐴, 𝐵⟩) = (℩𝑤⟨𝐴, 𝐵⟩uncurry 𝐹𝑤)
17	15, 16	eqtri 2768	. 2 ⊢ (𝐴uncurry 𝐹𝐵) = (℩𝑤⟨𝐴, 𝐵⟩uncurry 𝐹𝑤)
18		df-fv 6440	. 2 ⊢ ((𝐹‘𝐴)‘𝐵) = (℩𝑤𝐵(𝐹‘𝐴)𝑤)
19	14, 17, 18	3eqtr4g 2805	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴uncurry 𝐹𝐵) = ((𝐹‘𝐴)‘𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1542   ∈ wcel 2110  Vcvv 3431  ⟨cop 4573   class class class wbr 5079  ℩cio 6388  ‘cfv 6432  (class class class)co 7271  {coprab 7272  uncurry cunc 8073

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-iota 6390  df-fv 6440  df-ov 7274  df-oprab 7275  df-unc 8075

This theorem is referenced by:  curunc  35755  matunitlindflem2  35770