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Theorem mpteq12 3831
 Description: An equality theorem for the maps to notation. (Contributed by NM, 16-Dec-2013.)
Assertion
Ref Expression
mpteq12 ((A = 𝐶 x A B = 𝐷) → (x AB) = (x 𝐶𝐷))
Distinct variable groups:   x,A   x,𝐶
Allowed substitution hints:   B(x)   𝐷(x)

Proof of Theorem mpteq12
StepHypRef Expression
1 ax-17 1416 . 2 (A = 𝐶x A = 𝐶)
2 mpteq12f 3828 . 2 ((x A = 𝐶 x A B = 𝐷) → (x AB) = (x 𝐶𝐷))
31, 2sylan 267 1 ((A = 𝐶 x A B = 𝐷) → (x AB) = (x 𝐶𝐷))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1240   = wceq 1242  ∀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:  mpteq1  3832  mpteqb  5204  fmptcof  5274
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