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Theorem unceq 33875
 Description: Equality theorem for uncurrying. (Contributed by Brendan Leahy, 2-Jun-2021.)
Assertion
Ref Expression
unceq (𝐴 = 𝐵 → uncurry 𝐴 = uncurry 𝐵)

Proof of Theorem unceq
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq1 6410 . . . 4 (𝐴 = 𝐵 → (𝐴𝑥) = (𝐵𝑥))
21breqd 4854 . . 3 (𝐴 = 𝐵 → (𝑦(𝐴𝑥)𝑧𝑦(𝐵𝑥)𝑧))
32oprabbidv 6943 . 2 (𝐴 = 𝐵 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐴𝑥)𝑧} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐵𝑥)𝑧})
4 df-unc 7632 . 2 uncurry 𝐴 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐴𝑥)𝑧}
5 df-unc 7632 . 2 uncurry 𝐵 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐵𝑥)𝑧}
63, 4, 53eqtr4g 2858 1 (𝐴 = 𝐵 → uncurry 𝐴 = uncurry 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1653   class class class wbr 4843  ‘cfv 6101  {coprab 6879  uncurry cunc 7630 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777 This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-rex 3095  df-uni 4629  df-br 4844  df-iota 6064  df-fv 6109  df-oprab 6882  df-unc 7632 This theorem is referenced by: (None)
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