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Theorem resdifcom 4907
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 3372 . 2  |-  ( ( A  \  C )  i^i  ( B  X.  _V ) )  =  ( ( A  i^i  ( B  X.  _V ) ) 
\  C )
2 df-res 4621 . 2  |-  ( ( A  \  C )  |`  B )  =  ( ( A  \  C
)  i^i  ( B  X.  _V ) )
3 df-res 4621 . . 3  |-  ( A  |`  B )  =  ( A  i^i  ( B  X.  _V ) )
43difeq1i 3241 . 2  |-  ( ( A  |`  B )  \  C )  =  ( ( A  i^i  ( B  X.  _V ) ) 
\  C )
51, 2, 43eqtr4ri 2202 1  |-  ( ( A  |`  B )  \  C )  =  ( ( A  \  C
)  |`  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1348   _Vcvv 2730    \ cdif 3118    i^i cin 3120    X. cxp 4607    |` cres 4611
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rab 2457  df-v 2732  df-dif 3123  df-in 3127  df-res 4621
This theorem is referenced by:  setsfun0  12439
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