Theorem resdifcom 5056

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 3466	. 2 |- ( ( A \ C ) i^i ( B X. _V ) ) = ( ( A i^i ( B X. _V ) ) \ C )
2		df-res 4761	. 2 |- ( ( A \ C ) |` B ) = ( ( A \ C ) i^i ( B X. _V ) )
3		df-res 4761	. . 3 |- ( A |` B ) = ( A i^i ( B X. _V ) )
4	3	difeq1i 3333	. 2 |- ( ( A |` B ) \ C ) = ( ( A i^i ( B X. _V ) ) \ C )
5	1, 2, 4	3eqtr4ri 2264	1 |- ( ( A |` B ) \ C ) = ( ( A \ C ) |` B )

Syntax hints:   = wceq 1398   _Vcvv 2813   \ cdif 3208   i^i cin 3210   X. cxp 4747   |` cres 4751

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rab 2529  df-v 2815  df-dif 3213  df-in 3217  df-res 4761

This theorem is referenced by:  setsfun0  13248