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Theorem mpteq1 3841
 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 2041 . . 3 (𝑥𝐴𝐶 = 𝐶)
21rgen 2374 . 2 𝑥𝐴 𝐶 = 𝐶
3 mpteq12 3840 . 2 ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 𝐶 = 𝐶) → (𝑥𝐴𝐶) = (𝑥𝐵𝐶))
42, 3mpan2 401 1 (𝐴 = 𝐵 → (𝑥𝐴𝐶) = (𝑥𝐵𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243   ∈ wcel 1393  ∀wral 2306   ↦ cmpt 3818 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-ral 2311  df-opab 3819  df-mpt 3820 This theorem is referenced by:  mpteq1d  3842  fmptap  5353  mpt2mpt  5596  mpt2mptsx  5823  mpt2mpts  5824  tposf12  5884
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