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Theorem uncov 33370
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 4652 . . . . 5 (⟨𝐴, 𝐵⟩uncurry 𝐹𝑤 ↔ ⟨⟨𝐴, 𝐵⟩, 𝑤⟩ ∈ uncurry 𝐹)
2 df-unc 7391 . . . . . 6 uncurry 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹𝑥)𝑧}
32eleq2i 2692 . . . . 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 3201 . . . . 5 𝑤 ∈ V
6 simp2 1061 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝑤) → 𝑦 = 𝐵)
7 fveq2 6189 . . . . . . . 8 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
873ad2ant1 1081 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝑤) → (𝐹𝑥) = (𝐹𝐴))
9 simp3 1062 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝑤) → 𝑧 = 𝑤)
106, 8, 9breq123d 4665 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝑤) → (𝑦(𝐹𝑥)𝑧𝐵(𝐹𝐴)𝑤))
1110eloprabga 6744 . . . . 5 ((𝐴𝑉𝐵𝑊𝑤 ∈ V) → (⟨⟨𝐴, 𝐵⟩, 𝑤⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹𝑥)𝑧} ↔ 𝐵(𝐹𝐴)𝑤))
125, 11mp3an3 1412 . . . 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 5870 . 2 ((𝐴𝑉𝐵𝑊) → (℩𝑤𝐴, 𝐵⟩uncurry 𝐹𝑤) = (℩𝑤𝐵(𝐹𝐴)𝑤))
15 df-ov 6650 . . 3 (𝐴uncurry 𝐹𝐵) = (uncurry 𝐹‘⟨𝐴, 𝐵⟩)
16 df-fv 5894 . . 3 (uncurry 𝐹‘⟨𝐴, 𝐵⟩) = (℩𝑤𝐴, 𝐵⟩uncurry 𝐹𝑤)
1715, 16eqtri 2643 . 2 (𝐴uncurry 𝐹𝐵) = (℩𝑤𝐴, 𝐵⟩uncurry 𝐹𝑤)
18 df-fv 5894 . 2 ((𝐹𝐴)‘𝐵) = (℩𝑤𝐵(𝐹𝐴)𝑤)
1914, 17, 183eqtr4g 2680 1 ((𝐴𝑉𝐵𝑊) → (𝐴uncurry 𝐹𝐵) = ((𝐹𝐴)‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1037   = wceq 1482  wcel 1989  Vcvv 3198  cop 4181   class class class wbr 4651  cio 5847  cfv 5886  (class class class)co 6647  {coprab 6648  uncurry cunc 7389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-rex 2917  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-iota 5849  df-fv 5894  df-ov 6650  df-oprab 6651  df-unc 7391
This theorem is referenced by:  curunc  33371  matunitlindflem2  33386
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