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Theorem uncov 37588
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 5149 . . . . 5 (⟨𝐴, 𝐵⟩uncurry 𝐹𝑤 ↔ ⟨⟨𝐴, 𝐵⟩, 𝑤⟩ ∈ uncurry 𝐹)
2 df-unc 8292 . . . . . 6 uncurry 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹𝑥)𝑧}
32eleq2i 2831 . . . . 5 (⟨⟨𝐴, 𝐵⟩, 𝑤⟩ ∈ uncurry 𝐹 ↔ ⟨⟨𝐴, 𝐵⟩, 𝑤⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹𝑥)𝑧})
41, 3bitri 275 . . . 4 (⟨𝐴, 𝐵⟩uncurry 𝐹𝑤 ↔ ⟨⟨𝐴, 𝐵⟩, 𝑤⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹𝑥)𝑧})
5 vex 3482 . . . . 5 𝑤 ∈ V
6 simp2 1136 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝑤) → 𝑦 = 𝐵)
7 fveq2 6907 . . . . . . . 8 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
873ad2ant1 1132 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝑤) → (𝐹𝑥) = (𝐹𝐴))
9 simp3 1137 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝑤) → 𝑧 = 𝑤)
106, 8, 9breq123d 5162 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝑤) → (𝑦(𝐹𝑥)𝑧𝐵(𝐹𝐴)𝑤))
1110eloprabga 7541 . . . . 5 ((𝐴𝑉𝐵𝑊𝑤 ∈ V) → (⟨⟨𝐴, 𝐵⟩, 𝑤⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹𝑥)𝑧} ↔ 𝐵(𝐹𝐴)𝑤))
125, 11mp3an3 1449 . . . 4 ((𝐴𝑉𝐵𝑊) → (⟨⟨𝐴, 𝐵⟩, 𝑤⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹𝑥)𝑧} ↔ 𝐵(𝐹𝐴)𝑤))
134, 12bitrid 283 . . 3 ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩uncurry 𝐹𝑤𝐵(𝐹𝐴)𝑤))
1413iotabidv 6547 . 2 ((𝐴𝑉𝐵𝑊) → (℩𝑤𝐴, 𝐵⟩uncurry 𝐹𝑤) = (℩𝑤𝐵(𝐹𝐴)𝑤))
15 df-ov 7434 . . 3 (𝐴uncurry 𝐹𝐵) = (uncurry 𝐹‘⟨𝐴, 𝐵⟩)
16 df-fv 6571 . . 3 (uncurry 𝐹‘⟨𝐴, 𝐵⟩) = (℩𝑤𝐴, 𝐵⟩uncurry 𝐹𝑤)
1715, 16eqtri 2763 . 2 (𝐴uncurry 𝐹𝐵) = (℩𝑤𝐴, 𝐵⟩uncurry 𝐹𝑤)
18 df-fv 6571 . 2 ((𝐹𝐴)‘𝐵) = (℩𝑤𝐵(𝐹𝐴)𝑤)
1914, 17, 183eqtr4g 2800 1 ((𝐴𝑉𝐵𝑊) → (𝐴uncurry 𝐹𝐵) = ((𝐹𝐴)‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  Vcvv 3478  cop 4637   class class class wbr 5148  cio 6514  cfv 6563  (class class class)co 7431  {coprab 7432  uncurry cunc 8290
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-oprab 7435  df-unc 8292
This theorem is referenced by:  curunc  37589  matunitlindflem2  37604
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