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Theorem uncov 34314
Description: Value of uncurrying. (Contributed by Brendan Leahy, 2-Jun-2021.)
Assertion
Ref Expression
uncov ((𝐴𝑉𝐵𝑊) → (𝐴uncurry 𝐹𝐵) = ((𝐹𝐴)‘𝐵))

Proof of Theorem uncov
Dummy variables 𝑥 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-br 4924 . . . . 5 (⟨𝐴, 𝐵⟩uncurry 𝐹𝑤 ↔ ⟨⟨𝐴, 𝐵⟩, 𝑤⟩ ∈ uncurry 𝐹)
2 df-unc 7731 . . . . . 6 uncurry 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹𝑥)𝑧}
32eleq2i 2851 . . . . 5 (⟨⟨𝐴, 𝐵⟩, 𝑤⟩ ∈ uncurry 𝐹 ↔ ⟨⟨𝐴, 𝐵⟩, 𝑤⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹𝑥)𝑧})
41, 3bitri 267 . . . 4 (⟨𝐴, 𝐵⟩uncurry 𝐹𝑤 ↔ ⟨⟨𝐴, 𝐵⟩, 𝑤⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹𝑥)𝑧})
5 vex 3412 . . . . 5 𝑤 ∈ V
6 simp2 1117 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝑤) → 𝑦 = 𝐵)
7 fveq2 6493 . . . . . . . 8 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
873ad2ant1 1113 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝑤) → (𝐹𝑥) = (𝐹𝐴))
9 simp3 1118 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝑤) → 𝑧 = 𝑤)
106, 8, 9breq123d 4937 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝑤) → (𝑦(𝐹𝑥)𝑧𝐵(𝐹𝐴)𝑤))
1110eloprabga 7071 . . . . 5 ((𝐴𝑉𝐵𝑊𝑤 ∈ V) → (⟨⟨𝐴, 𝐵⟩, 𝑤⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹𝑥)𝑧} ↔ 𝐵(𝐹𝐴)𝑤))
125, 11mp3an3 1429 . . . 4 ((𝐴𝑉𝐵𝑊) → (⟨⟨𝐴, 𝐵⟩, 𝑤⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹𝑥)𝑧} ↔ 𝐵(𝐹𝐴)𝑤))
134, 12syl5bb 275 . . 3 ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩uncurry 𝐹𝑤𝐵(𝐹𝐴)𝑤))
1413iotabidv 6167 . 2 ((𝐴𝑉𝐵𝑊) → (℩𝑤𝐴, 𝐵⟩uncurry 𝐹𝑤) = (℩𝑤𝐵(𝐹𝐴)𝑤))
15 df-ov 6973 . . 3 (𝐴uncurry 𝐹𝐵) = (uncurry 𝐹‘⟨𝐴, 𝐵⟩)
16 df-fv 6190 . . 3 (uncurry 𝐹‘⟨𝐴, 𝐵⟩) = (℩𝑤𝐴, 𝐵⟩uncurry 𝐹𝑤)
1715, 16eqtri 2796 . 2 (𝐴uncurry 𝐹𝐵) = (℩𝑤𝐴, 𝐵⟩uncurry 𝐹𝑤)
18 df-fv 6190 . 2 ((𝐹𝐴)‘𝐵) = (℩𝑤𝐵(𝐹𝐴)𝑤)
1914, 17, 183eqtr4g 2833 1 ((𝐴𝑉𝐵𝑊) → (𝐴uncurry 𝐹𝐵) = ((𝐹𝐴)‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  w3a 1068   = wceq 1507  wcel 2050  Vcvv 3409  cop 4441   class class class wbr 4923  cio 6144  cfv 6182  (class class class)co 6970  {coprab 6971  uncurry cunc 7729
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2744  ax-sep 5054  ax-nul 5061  ax-pr 5180
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-rex 3088  df-rab 3091  df-v 3411  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-br 4924  df-iota 6146  df-fv 6190  df-ov 6973  df-oprab 6974  df-unc 7731
This theorem is referenced by:  curunc  34315  matunitlindflem2  34330
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