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Theorem mpteq1 4048
 Description: An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
Assertion
Ref Expression
mpteq1 (𝐴 = 𝐵 → (𝑥𝐴𝐶) = (𝑥𝐵𝐶))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem mpteq1
StepHypRef Expression
1 eqidd 2158 . . 3 (𝑥𝐴𝐶 = 𝐶)
21rgen 2510 . 2 𝑥𝐴 𝐶 = 𝐶
3 mpteq12 4047 . 2 ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 𝐶 = 𝐶) → (𝑥𝐴𝐶) = (𝑥𝐵𝐶))
42, 3mpan2 422 1 (𝐴 = 𝐵 → (𝑥𝐴𝐶) = (𝑥𝐵𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1335   ∈ wcel 2128  ∀wral 2435   ↦ cmpt 4025 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-ral 2440  df-opab 4026  df-mpt 4027 This theorem is referenced by:  mpteq1d  4049  fmptap  5654  mpompt  5907  mpomptsx  6139  mpompts  6140  tposf12  6210  restco  12534  cnmpt1t  12645  cnmpt2t  12653
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