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Theorem resmpti 44430
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 6030 . 2 (𝐵𝐴 → ((𝑥𝐴𝐶) ↾ 𝐵) = (𝑥𝐵𝐶))
31, 2ax-mp 5 1 ((𝑥𝐴𝐶) ↾ 𝐵) = (𝑥𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wss 3943  cmpt 5224  cres 5671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-opab 5204  df-mpt 5225  df-xp 5675  df-rel 5676  df-res 5681
This theorem is referenced by:  sge0splitmpt  45680
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