Theorem mpteq1i 5243

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 2735	. . 3 ⊢ (⊤ → 𝐶 = 𝐶)
4	2, 3	mpteq12dv 5238	. 2 ⊢ (⊤ → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶))
5	4	mptru 1543	1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶)

Syntax hints:   = wceq 1536  ⊤wtru 1537   ↦ cmpt 5230

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-opab 5210  df-mpt 5231

This theorem is referenced by:  fmptap  7189  mpompt  7546  offres  8006  mpomptsx  8087  mpompts  8088  pwfseq  10701  wrd2f1tovbij  14995  pmtrprfval  19519  gsum2dlem2  20003  gsumcom2  20007  srgbinomlem4  20246  ply1coe  22317  m2detleiblem3  22650  m2detleiblem4  22651  pmatcollpw3fi1lem1  22807  restco  23187  limcdif  25925  dfarea  27017  nosupcbv  27761  noinfcbv  27776  istrkg2ld  28482  wlknwwlksnbij  29917  wwlksnextbij  29931  clwlknf1oclwwlkn  30112  dfhnorm2  31150  ccatws1f1o  32920  gsumwrd2dccat  33052  algextdeglem4  33725  algextdeglem5  33726  trlset  40143  limsupequzmptlem  45683  sge0iunmptlemfi  46368  sge0iunmpt  46373  hoidmvlelem3  46552  smfmulc1  46751  smflimsuplem2  46776