Users' Mathboxes Mathbox for Brendan Leahy < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  unceq Structured version   Visualization version   GIF version

Theorem unceq 37584
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 6906 . . . 4 (𝐴 = 𝐵 → (𝐴𝑥) = (𝐵𝑥))
21breqd 5159 . . 3 (𝐴 = 𝐵 → (𝑦(𝐴𝑥)𝑧𝑦(𝐵𝑥)𝑧))
32oprabbidv 7499 . 2 (𝐴 = 𝐵 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐴𝑥)𝑧} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐵𝑥)𝑧})
4 df-unc 8292 . 2 uncurry 𝐴 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐴𝑥)𝑧}
5 df-unc 8292 . 2 uncurry 𝐵 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐵𝑥)𝑧}
63, 4, 53eqtr4g 2800 1 (𝐴 = 𝐵 → uncurry 𝐴 = uncurry 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537   class class class wbr 5148  cfv 6563  {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-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-ss 3980  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-oprab 7435  df-unc 8292
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator