Theorem resdifcom 4909

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

Step	Hyp	Ref	Expression
1		indif1 3372	. 2 |- ( ( A \ C ) i^i ( B X. _V ) ) = ( ( A i^i ( B X. _V ) ) \ C )
2		df-res 4623	. 2 |- ( ( A \ C ) |` B ) = ( ( A \ C ) i^i ( B X. _V ) )
3		df-res 4623	. . 3 |- ( A |` B ) = ( A i^i ( B X. _V ) )
4	3	difeq1i 3241	. 2 |- ( ( A |` B ) \ C ) = ( ( A i^i ( B X. _V ) ) \ C )
5	1, 2, 4	3eqtr4ri 2202	1 |- ( ( A |` B ) \ C ) = ( ( A \ C ) |` B )

Syntax hints:   = wceq 1348   _Vcvv 2730   \ cdif 3118   i^i cin 3120   X. cxp 4609   |` cres 4613

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 4623

This theorem is referenced by:  setsfun0  12452