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

Step	Hyp	Ref	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 ) )
4	3	difeq1i 3264	. 2 |- ( ( A |` B ) \ C ) = ( ( A i^i ( B X. _V ) ) \ C )
5	1, 2, 4	3eqtr4ri 2221	1 |- ( ( A |` B ) \ C ) = ( ( A \ C ) |` B )

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