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Theorem mpteq1 3972
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 2116 . . 3 (𝑥𝐴𝐶 = 𝐶)
21rgen 2459 . 2 𝑥𝐴 𝐶 = 𝐶
3 mpteq12 3971 . 2 ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 𝐶 = 𝐶) → (𝑥𝐴𝐶) = (𝑥𝐵𝐶))
42, 3mpan2 419 1 (𝐴 = 𝐵 → (𝑥𝐴𝐶) = (𝑥𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1314  wcel 1463  wral 2390  cmpt 3949
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-11 1467  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-ral 2395  df-opab 3950  df-mpt 3951
This theorem is referenced by:  mpteq1d  3973  fmptap  5564  mpompt  5817  mpomptsx  6049  mpompts  6050  tposf12  6120  restco  12186  cnmpt1t  12296  cnmpt2t  12304
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