Theorem inres2 34007

Description: Two ways of expressing the restriction of an intersection. (Contributed by Peter Mazsa, 5-Jun-2021.)

Assertion

Ref	Expression
inres2	|- ( ( R |` A ) i^i S ) = ( ( R i^i S ) |` A )

Proof of Theorem inres2

Step	Hyp	Ref	Expression
1		inres 5414	. . 3 |- ( S i^i ( R |` A ) ) = ( ( S i^i R ) |` A )
2	1	ineqcomi 34006	. 2 |- ( ( R |` A ) i^i S ) = ( ( S i^i R ) |` A )
3		incom 3805	. . 3 |- ( R i^i S ) = ( S i^i R )
4	3	reseq1i 5392	. 2 |- ( ( R i^i S ) |` A ) = ( ( S i^i R ) |` A )
5	2, 4	eqtr4i 2647	1 |- ( ( R |` A ) i^i S ) = ( ( R i^i S ) |` A )

Syntax hints:   = wceq 1483   i^i cin 3573   |` 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-v 3202  df-in 3581  df-res 5126

This theorem is referenced by: (None)