Theorem mpteq1i 5149

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 5147	. 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 1533   ↦ cmpt 5139

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-12 2172  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-ral 3143  df-opab 5122  df-mpt 5140

This theorem is referenced by:  fmptap  6927  mpompt  7260  offres  7678  mpomptsx  7756  mpompts  7757  pwfseq  10080  wrd2f1tovbij  14318  pmtrprfval  18609  gsum2dlem2  19085  gsumcom2  19089  srgbinomlem4  19287  ply1coe  20458  m2detleiblem3  21232  m2detleiblem4  21233  pmatcollpw3fi1lem1  21388  restco  21766  limcdif  24468  dfarea  25532  istrkg2ld  26240  wlknwwlksnbij  27660  wwlksnextbij  27674  clwlknf1oclwwlkn  27857  dfhnorm2  28893  trlset  37291  limsupequzmptlem  42001  sge0iunmptlemfi  42688  sge0iunmpt  42693  hoidmvlelem3  42872  smfmulc1  43064  smflimsuplem2  43088