Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  mpteq1i Structured version   Visualization version   GIF version

Theorem mpteq1i 4873
 Description: An equality theorem for the maps to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypothesis
Ref Expression
mpteq1i.1 𝐴 = 𝐵
Assertion
Ref Expression
mpteq1i (𝑥𝐴𝐶) = (𝑥𝐵𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem mpteq1i
StepHypRef Expression
1 mpteq1i.1 . 2 𝐴 = 𝐵
2 mpteq1 4871 . 2 (𝐴 = 𝐵 → (𝑥𝐴𝐶) = (𝑥𝐵𝐶))
31, 2ax-mp 5 1 (𝑥𝐴𝐶) = (𝑥𝐵𝐶)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1631   ↦ cmpt 4863 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-ral 3066  df-opab 4847  df-mpt 4864 This theorem is referenced by:  wlknwwlksnbij  27026  wlkwwlkbij2OLD  27034  wwlksnextbij  27046  clwlknf1oclwwlkn  27255  limsupequzmptlem  40478  sge0iunmptlemfi  41147  sge0iunmpt  41152  hoidmvlelem3  41331  smfmulc1  41523  smflimsuplem2  41547
 Copyright terms: Public domain W3C validator