Theorem resdifcom 5415

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 3871	. 2 |- ( ( A \ C ) i^i ( B X. _V ) ) = ( ( A i^i ( B X. _V ) ) \ C )
2		df-res 5126	. 2 |- ( ( A \ C ) |` B ) = ( ( A \ C ) i^i ( B X. _V ) )
3		df-res 5126	. . 3 |- ( A |` B ) = ( A i^i ( B X. _V ) )
4	3	difeq1i 3724	. 2 |- ( ( A |` B ) \ C ) = ( ( A i^i ( B X. _V ) ) \ C )
5	1, 2, 4	3eqtr4ri 2655	1 |- ( ( A |` B ) \ C ) = ( ( A \ C ) |` B )

Syntax hints:   = wceq 1483   _Vcvv 3200   \ cdif 3571   i^i cin 3573   X. cxp 5112   |` cres 5116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-in 3581  df-res 5126

This theorem is referenced by:  setsfun0  15894