MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mpteq12i Structured version   Visualization version   GIF version

Theorem mpteq12i 4704
Description: An equality inference for the maps to notation. (Contributed by Scott Fenton, 27-Oct-2010.) (Revised by Mario Carneiro, 16-Dec-2013.)
Hypotheses
Ref Expression
mpteq12i.1 𝐴 = 𝐶
mpteq12i.2 𝐵 = 𝐷
Assertion
Ref Expression
mpteq12i (𝑥𝐴𝐵) = (𝑥𝐶𝐷)

Proof of Theorem mpteq12i
StepHypRef Expression
1 mpteq12i.1 . . . 4 𝐴 = 𝐶
21a1i 11 . . 3 (⊤ → 𝐴 = 𝐶)
3 mpteq12i.2 . . . 4 𝐵 = 𝐷
43a1i 11 . . 3 (⊤ → 𝐵 = 𝐷)
52, 4mpteq12dv 4695 . 2 (⊤ → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
65trud 1490 1 (𝑥𝐴𝐵) = (𝑥𝐶𝐷)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  wtru 1481  cmpt 4675
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-ral 2912  df-opab 4676  df-mpt 4677
This theorem is referenced by:  offres  7111  pmtrprfval  17831  evlsval  19441  madufval  20365  limcdif  23553  dfhnorm2  27840  cdj3lem3  29158  cdj3lem3b  29160  partfun  29330  esumsnf  29919  esumrnmpt2  29923  measinb2  30079  eulerpart  30237  fiblem  30253  trlset  34949  hoidmvlelem4  40135  smflimsup  40357
  Copyright terms: Public domain W3C validator