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Theorem mpteq1 3832
 Description: An equality theorem for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
Assertion
Ref Expression
mpteq1 (A = B → (x A𝐶) = (x B𝐶))
Distinct variable groups:   x,A   x,B
Allowed substitution hint:   𝐶(x)

Proof of Theorem mpteq1
StepHypRef Expression
1 eqidd 2038 . . 3 (x A𝐶 = 𝐶)
21rgen 2368 . 2 x A 𝐶 = 𝐶
3 mpteq12 3831 . 2 ((A = B x A 𝐶 = 𝐶) → (x A𝐶) = (x B𝐶))
42, 3mpan2 401 1 (A = B → (x A𝐶) = (x B𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1242   ∈ wcel 1390  ∀wral 2300   ↦ cmpt 3809 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 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-ral 2305  df-opab 3810  df-mpt 3811 This theorem is referenced by:  mpteq1d  3833  fmptap  5296  mpt2mpt  5538  mpt2mptsx  5765  mpt2mpts  5766  tposf12  5825
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