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Theorem resabs1d 4861
 Description: Absorption law for restriction, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypothesis
Ref Expression
resabs1d.b (𝜑𝐵𝐶)
Assertion
Ref Expression
resabs1d (𝜑 → ((𝐴𝐶) ↾ 𝐵) = (𝐴𝐵))

Proof of Theorem resabs1d
StepHypRef Expression
1 resabs1d.b . 2 (𝜑𝐵𝐶)
2 resabs1 4860 . 2 (𝐵𝐶 → ((𝐴𝐶) ↾ 𝐵) = (𝐴𝐵))
31, 2syl 14 1 (𝜑 → ((𝐴𝐶) ↾ 𝐵) = (𝐴𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1332   ⊆ wss 3078   ↾ 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-xp 4557  df-rel 4558  df-res 4563 This theorem is referenced by:  resubmet  12792
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