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Theorem mpteq12df 4705
Description: An equality theorem for the maps to notation. (Contributed by Thierry Arnoux, 30-May-2020.)
Hypotheses
Ref Expression
mpteq12df.0 𝑥𝜑
mpteq12df.1 𝑥𝐴
mpteq12df.2 𝑥𝐶
mpteq12df.3 (𝜑𝐴 = 𝐶)
mpteq12df.4 (𝜑𝐵 = 𝐷)
Assertion
Ref Expression
mpteq12df (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))

Proof of Theorem mpteq12df
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 mpteq12df.0 . . 3 𝑥𝜑
2 nfv 1840 . . 3 𝑦𝜑
3 mpteq12df.3 . . . . 5 (𝜑𝐴 = 𝐶)
43eleq2d 2684 . . . 4 (𝜑 → (𝑥𝐴𝑥𝐶))
5 mpteq12df.4 . . . . 5 (𝜑𝐵 = 𝐷)
65eqeq2d 2631 . . . 4 (𝜑 → (𝑦 = 𝐵𝑦 = 𝐷))
74, 6anbi12d 746 . . 3 (𝜑 → ((𝑥𝐴𝑦 = 𝐵) ↔ (𝑥𝐶𝑦 = 𝐷)))
81, 2, 7opabbid 4687 . 2 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦 = 𝐷)})
9 df-mpt 4685 . 2 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
10 df-mpt 4685 . 2 (𝑥𝐶𝐷) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦 = 𝐷)}
118, 9, 103eqtr4g 2680 1 (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wnf 1705  wcel 1987  wnfc 2748  {copab 4682  cmpt 4683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-opab 4684  df-mpt 4685
This theorem is referenced by:  esumrnmpt2  29953  smflimsuplem3  40365
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