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Theorem resdifcom 4837
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 3321 . 2  |-  ( ( A  \  C )  i^i  ( B  X.  _V ) )  =  ( ( A  i^i  ( B  X.  _V ) ) 
\  C )
2 df-res 4551 . 2  |-  ( ( A  \  C )  |`  B )  =  ( ( A  \  C
)  i^i  ( B  X.  _V ) )
3 df-res 4551 . . 3  |-  ( A  |`  B )  =  ( A  i^i  ( B  X.  _V ) )
43difeq1i 3190 . 2  |-  ( ( A  |`  B )  \  C )  =  ( ( A  i^i  ( B  X.  _V ) ) 
\  C )
51, 2, 43eqtr4ri 2171 1  |-  ( ( A  |`  B )  \  C )  =  ( ( A  \  C
)  |`  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1331   _Vcvv 2686    \ cdif 3068    i^i cin 3070    X. cxp 4537    |` cres 4541
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-in1 603  ax-in2 604  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-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rab 2425  df-v 2688  df-dif 3073  df-in 3077  df-res 4551
This theorem is referenced by:  setsfun0  12009
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