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Theorem mpteq1d 4100
Description: An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 11-Jun-2016.)
Hypothesis
Ref Expression
mpteq1d.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
mpteq1d (𝜑 → (𝑥𝐴𝐶) = (𝑥𝐵𝐶))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝐶(𝑥)

Proof of Theorem mpteq1d
StepHypRef Expression
1 mpteq1d.1 . 2 (𝜑𝐴 = 𝐵)
2 mpteq1 4099 . 2 (𝐴 = 𝐵 → (𝑥𝐴𝐶) = (𝑥𝐵𝐶))
31, 2syl 14 1 (𝜑 → (𝑥𝐴𝐶) = (𝑥𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1363  cmpt 4076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-11 1516  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-ral 2470  df-opab 4077  df-mpt 4078
This theorem is referenced by:  fmptapd  5720  offval  6104  qusex  12764
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