Theorem mpteq1i 5177

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 2738	. . 3 ⊢ (⊤ → 𝐶 = 𝐶)
4	2, 3	mpteq12dv 5173	. 2 ⊢ (⊤ → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶))
5	4	mptru 1549	1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶)

Syntax hints:   = wceq 1542  ⊤wtru 1543   ↦ cmpt 5167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-opab 5149  df-mpt 5168

This theorem is referenced by:  fmptap  7116  mpompt  7472  offres  7927  mpomptsx  8008  mpompts  8009  pwfseq  10576  wrd2f1tovbij  14884  pmtrprfval  19420  gsum2dlem2  19904  gsumcom2  19908  srgbinomlem4  20168  ply1coe  22241  m2detleiblem3  22572  m2detleiblem4  22573  pmatcollpw3fi1lem1  22729  restco  23107  limcdif  25821  dfarea  26910  nosupcbv  27654  noinfcbv  27669  istrkg2ld  28516  wlknwwlksnbij  29945  wwlksnextbij  29959  clwlknf1oclwwlkn  30143  dfhnorm2  31182  partfun2  32738  ccatws1f1o  33016  gsumwrd2dccat  33144  vietalem  33728  algextdeglem4  33870  algextdeglem5  33871  dfadjliftmap2  38769  dfblockliftmap2  38773  trlset  40598  limsupequzmptlem  46160  sge0iunmptlemfi  46845  sge0iunmpt  46850  hoidmvlelem3  47029  smfmulc1  47228  smflimsuplem2  47253  tposrescnv  49312  swapf1f1o  49708  precofval3  49804