Theorem mpteq1i 5166

Description: An equality theorem for the maps-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.) Remove all disjoint variable conditions. (Revised by SN, 11-Nov-2024.)

Hypothesis

Ref	Expression
mpteq1i.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
mpteq1i	⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶)

Proof of Theorem mpteq1i

Step	Hyp	Ref	Expression
1		mpteq1i.1	. . . 4 ⊢ 𝐴 = 𝐵
2	1	a1i 11	. . 3 ⊢ (⊤ → 𝐴 = 𝐵)
3		eqidd 2739	. . 3 ⊢ (⊤ → 𝐶 = 𝐶)
4	2, 3	mpteq12dv 5161	. 2 ⊢ (⊤ → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶))
5	4	mptru 1546	1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶)

Syntax hints:   = wceq 1539  ⊤wtru 1540   ↦ cmpt 5153

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-opab 5133  df-mpt 5154

This theorem is referenced by:  fmptap  7024  mpompt  7366  offres  7799  mpomptsx  7877  mpompts  7878  pwfseq  10351  wrd2f1tovbij  14603  pmtrprfval  19010  gsum2dlem2  19487  gsumcom2  19491  srgbinomlem4  19694  ply1coe  21377  m2detleiblem3  21686  m2detleiblem4  21687  pmatcollpw3fi1lem1  21843  restco  22223  limcdif  24945  dfarea  26015  istrkg2ld  26725  wlknwwlksnbij  28154  wwlksnextbij  28168  clwlknf1oclwwlkn  28349  dfhnorm2  29385  nosupcbv  33832  noinfcbv  33847  trlset  38102  limsupequzmptlem  43159  sge0iunmptlemfi  43841  sge0iunmpt  43846  hoidmvlelem3  44025  smfmulc1  44217  smflimsuplem2  44241