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Theorem uncov 34754
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 5058 . . . . 5 (⟨𝐴, 𝐵⟩uncurry 𝐹𝑤 ↔ ⟨⟨𝐴, 𝐵⟩, 𝑤⟩ ∈ uncurry 𝐹)
2 df-unc 7923 . . . . . 6 uncurry 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹𝑥)𝑧}
32eleq2i 2901 . . . . 5 (⟨⟨𝐴, 𝐵⟩, 𝑤⟩ ∈ uncurry 𝐹 ↔ ⟨⟨𝐴, 𝐵⟩, 𝑤⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹𝑥)𝑧})
41, 3bitri 276 . . . 4 (⟨𝐴, 𝐵⟩uncurry 𝐹𝑤 ↔ ⟨⟨𝐴, 𝐵⟩, 𝑤⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹𝑥)𝑧})
5 vex 3495 . . . . 5 𝑤 ∈ V
6 simp2 1129 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝑤) → 𝑦 = 𝐵)
7 fveq2 6663 . . . . . . . 8 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
873ad2ant1 1125 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝑤) → (𝐹𝑥) = (𝐹𝐴))
9 simp3 1130 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝑤) → 𝑧 = 𝑤)
106, 8, 9breq123d 5071 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝑤) → (𝑦(𝐹𝑥)𝑧𝐵(𝐹𝐴)𝑤))
1110eloprabga 7250 . . . . 5 ((𝐴𝑉𝐵𝑊𝑤 ∈ V) → (⟨⟨𝐴, 𝐵⟩, 𝑤⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹𝑥)𝑧} ↔ 𝐵(𝐹𝐴)𝑤))
125, 11mp3an3 1441 . . . 4 ((𝐴𝑉𝐵𝑊) → (⟨⟨𝐴, 𝐵⟩, 𝑤⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹𝑥)𝑧} ↔ 𝐵(𝐹𝐴)𝑤))
134, 12syl5bb 284 . . 3 ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩uncurry 𝐹𝑤𝐵(𝐹𝐴)𝑤))
1413iotabidv 6332 . 2 ((𝐴𝑉𝐵𝑊) → (℩𝑤𝐴, 𝐵⟩uncurry 𝐹𝑤) = (℩𝑤𝐵(𝐹𝐴)𝑤))
15 df-ov 7148 . . 3 (𝐴uncurry 𝐹𝐵) = (uncurry 𝐹‘⟨𝐴, 𝐵⟩)
16 df-fv 6356 . . 3 (uncurry 𝐹‘⟨𝐴, 𝐵⟩) = (℩𝑤𝐴, 𝐵⟩uncurry 𝐹𝑤)
1715, 16eqtri 2841 . 2 (𝐴uncurry 𝐹𝐵) = (℩𝑤𝐴, 𝐵⟩uncurry 𝐹𝑤)
18 df-fv 6356 . 2 ((𝐹𝐴)‘𝐵) = (℩𝑤𝐵(𝐹𝐴)𝑤)
1914, 17, 183eqtr4g 2878 1 ((𝐴𝑉𝐵𝑊) → (𝐴uncurry 𝐹𝐵) = ((𝐹𝐴)‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  Vcvv 3492  cop 4563   class class class wbr 5057  cio 6305  cfv 6348  (class class class)co 7145  {coprab 7146  uncurry cunc 7921
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-ov 7148  df-oprab 7149  df-unc 7923
This theorem is referenced by:  curunc  34755  matunitlindflem2  34770
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