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Theorem resmptd 4882
 Description: Restriction of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypothesis
Ref Expression
resmptd.b (𝜑𝐵𝐴)
Assertion
Ref Expression
resmptd (𝜑 → ((𝑥𝐴𝐶) ↾ 𝐵) = (𝑥𝐵𝐶))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝐶(𝑥)

Proof of Theorem resmptd
StepHypRef Expression
1 resmptd.b . 2 (𝜑𝐵𝐴)
2 resmpt 4879 . 2 (𝐵𝐴 → ((𝑥𝐴𝐶) ↾ 𝐵) = (𝑥𝐵𝐶))
31, 2syl 14 1 (𝜑 → ((𝑥𝐴𝐶) ↾ 𝐵) = (𝑥𝐵𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1332   ⊆ wss 3078   ↦ cmpt 3999   ↾ cres 4553 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-14 2115  ax-ext 2123  ax-sep 4056  ax-pow 4108  ax-pr 4142 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1732  df-clab 2128  df-cleq 2134  df-clel 2137  df-nfc 2272  df-ral 2423  df-rex 2424  df-v 2693  df-un 3082  df-in 3084  df-ss 3091  df-pw 3519  df-sn 3540  df-pr 3541  df-op 3543  df-opab 4000  df-mpt 4001  df-xp 4557  df-rel 4558  df-res 4563 This theorem is referenced by:  fisumss  11222  fprodssdc  11420  cnmpt1res  12540
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