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Theorem uncov 33723
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 4787 . . . . 5 (⟨𝐴, 𝐵⟩uncurry 𝐹𝑤 ↔ ⟨⟨𝐴, 𝐵⟩, 𝑤⟩ ∈ uncurry 𝐹)
2 df-unc 7546 . . . . . 6 uncurry 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹𝑥)𝑧}
32eleq2i 2842 . . . . 5 (⟨⟨𝐴, 𝐵⟩, 𝑤⟩ ∈ uncurry 𝐹 ↔ ⟨⟨𝐴, 𝐵⟩, 𝑤⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹𝑥)𝑧})
41, 3bitri 264 . . . 4 (⟨𝐴, 𝐵⟩uncurry 𝐹𝑤 ↔ ⟨⟨𝐴, 𝐵⟩, 𝑤⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹𝑥)𝑧})
5 vex 3354 . . . . 5 𝑤 ∈ V
6 simp2 1131 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝑤) → 𝑦 = 𝐵)
7 fveq2 6332 . . . . . . . 8 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
873ad2ant1 1127 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝑤) → (𝐹𝑥) = (𝐹𝐴))
9 simp3 1132 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝑤) → 𝑧 = 𝑤)
106, 8, 9breq123d 4800 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝑤) → (𝑦(𝐹𝑥)𝑧𝐵(𝐹𝐴)𝑤))
1110eloprabga 6894 . . . . 5 ((𝐴𝑉𝐵𝑊𝑤 ∈ V) → (⟨⟨𝐴, 𝐵⟩, 𝑤⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹𝑥)𝑧} ↔ 𝐵(𝐹𝐴)𝑤))
125, 11mp3an3 1561 . . . 4 ((𝐴𝑉𝐵𝑊) → (⟨⟨𝐴, 𝐵⟩, 𝑤⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹𝑥)𝑧} ↔ 𝐵(𝐹𝐴)𝑤))
134, 12syl5bb 272 . . 3 ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩uncurry 𝐹𝑤𝐵(𝐹𝐴)𝑤))
1413iotabidv 6015 . 2 ((𝐴𝑉𝐵𝑊) → (℩𝑤𝐴, 𝐵⟩uncurry 𝐹𝑤) = (℩𝑤𝐵(𝐹𝐴)𝑤))
15 df-ov 6796 . . 3 (𝐴uncurry 𝐹𝐵) = (uncurry 𝐹‘⟨𝐴, 𝐵⟩)
16 df-fv 6039 . . 3 (uncurry 𝐹‘⟨𝐴, 𝐵⟩) = (℩𝑤𝐴, 𝐵⟩uncurry 𝐹𝑤)
1715, 16eqtri 2793 . 2 (𝐴uncurry 𝐹𝐵) = (℩𝑤𝐴, 𝐵⟩uncurry 𝐹𝑤)
18 df-fv 6039 . 2 ((𝐹𝐴)‘𝐵) = (℩𝑤𝐵(𝐹𝐴)𝑤)
1914, 17, 183eqtr4g 2830 1 ((𝐴𝑉𝐵𝑊) → (𝐴uncurry 𝐹𝐵) = ((𝐹𝐴)‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  Vcvv 3351  cop 4322   class class class wbr 4786  cio 5992  cfv 6031  (class class class)co 6793  {coprab 6794  uncurry cunc 7544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-iota 5994  df-fv 6039  df-ov 6796  df-oprab 6797  df-unc 7546
This theorem is referenced by:  curunc  33724  matunitlindflem2  33739
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