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Theorem resmpti 41817
 Description: Restriction of the mapping operation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypothesis
Ref Expression
resmpti.1 𝐵𝐴
Assertion
Ref Expression
resmpti ((𝑥𝐴𝐶) ↾ 𝐵) = (𝑥𝐵𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem resmpti
StepHypRef Expression
1 resmpti.1 . 2 𝐵𝐴
2 resmpt 5872 . 2 (𝐵𝐴 → ((𝑥𝐴𝐶) ↾ 𝐵) = (𝑥𝐵𝐶))
31, 2ax-mp 5 1 ((𝑥𝐴𝐶) ↾ 𝐵) = (𝑥𝐵𝐶)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ⊆ wss 3881   ↦ cmpt 5110   ↾ cres 5521 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-opab 5093  df-mpt 5111  df-xp 5525  df-rel 5526  df-res 5531 This theorem is referenced by:  sge0splitmpt  43065
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