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Theorem resmpti 42387
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 5905 . 2 (𝐵𝐴 → ((𝑥𝐴𝐶) ↾ 𝐵) = (𝑥𝐵𝐶))
31, 2ax-mp 5 1 ((𝑥𝐴𝐶) ↾ 𝐵) = (𝑥𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1543  wss 3866  cmpt 5135  cres 5553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-opab 5116  df-mpt 5136  df-xp 5557  df-rel 5558  df-res 5563
This theorem is referenced by:  sge0splitmpt  43624
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