MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mpt2eq3ia Structured version   Visualization version   GIF version

Theorem mpt2eq3ia 6673
Description: An equality inference for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
Hypothesis
Ref Expression
mpt2eq3ia.1 ((𝑥𝐴𝑦𝐵) → 𝐶 = 𝐷)
Assertion
Ref Expression
mpt2eq3ia (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐴, 𝑦𝐵𝐷)

Proof of Theorem mpt2eq3ia
StepHypRef Expression
1 mpt2eq3ia.1 . . . 4 ((𝑥𝐴𝑦𝐵) → 𝐶 = 𝐷)
213adant1 1077 . . 3 ((⊤ ∧ 𝑥𝐴𝑦𝐵) → 𝐶 = 𝐷)
32mpt2eq3dva 6672 . 2 (⊤ → (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐴, 𝑦𝐵𝐷))
43trud 1490 1 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐴, 𝑦𝐵𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wtru 1481  wcel 1987  cmpt2 6606
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-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-oprab 6608  df-mpt2 6609
This theorem is referenced by:  mpt2difsnif  6706  mpt2snif  6707  oprab2co  7207  cnfcomlem  8540  cnfcom2  8543  dfioo2  12216  elovmpt2wrd  13286  sadcom  15109  comfffval2  16282  oppchomf  16301  symgga  17747  oppglsm  17978  dfrhm2  18638  cnfldsub  19693  cnflddiv  19695  mat0op  20144  mattpos1  20181  mdetunilem7  20343  madufval  20362  maducoeval2  20365  madugsum  20368  mp2pm2mplem5  20534  mp2pm2mp  20535  leordtval  20927  xpstopnlem1  21522  divcn  22579  oprpiece1res1  22658  oprpiece1res2  22659  cxpcn  24386  numclwwlk5  27100  numclwwlk6  27102  cnnvm  27383  mdetpmtr2  29669  madjusmdetlem1  29672  cnre2csqima  29736  mndpluscn  29751  raddcn  29754  icorempt2  32828  matunitlindflem1  33034  mendplusgfval  37233  hoidmv1le  40112  hspdifhsp  40134  vonn0ioo  40205  vonn0icc  40206  dflinc2  41484
  Copyright terms: Public domain W3C validator