Theorem mpteq1i 5159

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	⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem mpteq1i

Step	Hyp	Ref	Expression
1		mpteq1i.1	. 2 ⊢ 𝐴 = 𝐵
2		mpteq1 5157	. 2 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶))
3	1, 2	ax-mp 5	1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶)

Syntax hints:   = wceq 1536   ↦ cmpt 5149

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-12 2176  ax-ext 2796

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-ral 3146  df-opab 5132  df-mpt 5150

This theorem is referenced by:  fmptap  6935  mpompt  7269  offres  7687  mpomptsx  7765  mpompts  7766  pwfseq  10089  wrd2f1tovbij  14327  pmtrprfval  18618  gsum2dlem2  19094  gsumcom2  19098  srgbinomlem4  19296  ply1coe  20467  m2detleiblem3  21241  m2detleiblem4  21242  pmatcollpw3fi1lem1  21397  restco  21775  limcdif  24477  dfarea  25541  istrkg2ld  26249  wlknwwlksnbij  27669  wwlksnextbij  27683  clwlknf1oclwwlkn  27866  dfhnorm2  28902  trlset  37301  limsupequzmptlem  42015  sge0iunmptlemfi  42702  sge0iunmpt  42707  hoidmvlelem3  42886  smfmulc1  43078  smflimsuplem2  43102