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Theorem resdifcom 4943
Description: Commutative law for restriction and difference. (Contributed by AV, 7-Jun-2021.)
Assertion
Ref Expression
resdifcom  |-  ( ( A  |`  B )  \  C )  =  ( ( A  \  C
)  |`  B )

Proof of Theorem resdifcom
StepHypRef Expression
1 indif1 3395 . 2  |-  ( ( A  \  C )  i^i  ( B  X.  _V ) )  =  ( ( A  i^i  ( B  X.  _V ) ) 
\  C )
2 df-res 4656 . 2  |-  ( ( A  \  C )  |`  B )  =  ( ( A  \  C
)  i^i  ( B  X.  _V ) )
3 df-res 4656 . . 3  |-  ( A  |`  B )  =  ( A  i^i  ( B  X.  _V ) )
43difeq1i 3264 . 2  |-  ( ( A  |`  B )  \  C )  =  ( ( A  i^i  ( B  X.  _V ) ) 
\  C )
51, 2, 43eqtr4ri 2221 1  |-  ( ( A  |`  B )  \  C )  =  ( ( A  \  C
)  |`  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1364   _Vcvv 2752    \ cdif 3141    i^i cin 3143    X. cxp 4642    |` cres 4646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rab 2477  df-v 2754  df-dif 3146  df-in 3150  df-res 4656
This theorem is referenced by:  setsfun0  12548
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