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Theorem resabs1i 39835
 Description: Absorption law for restriction. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
Hypothesis
Ref Expression
resabs1i.1 𝐵𝐶
Assertion
Ref Expression
resabs1i ((𝐴𝐶) ↾ 𝐵) = (𝐴𝐵)

Proof of Theorem resabs1i
StepHypRef Expression
1 resabs1i.1 . 2 𝐵𝐶
2 resabs1 5585 . 2 (𝐵𝐶 → ((𝐴𝐶) ↾ 𝐵) = (𝐴𝐵))
31, 2ax-mp 5 1 ((𝐴𝐶) ↾ 𝐵) = (𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1632   ⊆ wss 3715   ↾ cres 5268 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-opab 4865  df-xp 5272  df-rel 5273  df-res 5278 This theorem is referenced by:  liminfresre  40514
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