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Theorem resabs1d 4844
Description: Absorption law for restriction, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypothesis
Ref Expression
resabs1d.b  |-  ( ph  ->  B  C_  C )
Assertion
Ref Expression
resabs1d  |-  ( ph  ->  ( ( A  |`  C )  |`  B )  =  ( A  |`  B ) )

Proof of Theorem resabs1d
StepHypRef Expression
1 resabs1d.b . 2  |-  ( ph  ->  B  C_  C )
2 resabs1 4843 . 2  |-  ( B 
C_  C  ->  (
( A  |`  C )  |`  B )  =  ( A  |`  B )
)
31, 2syl 14 1  |-  ( ph  ->  ( ( A  |`  C )  |`  B )  =  ( A  |`  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331    C_ wss 3066    |` cres 4536
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-opab 3985  df-xp 4540  df-rel 4541  df-res 4546
This theorem is referenced by:  resubmet  12706
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