Theorem inxpssres 34076

Description: Intersection with a Cartesian product is a subclass of restriction. (Contributed by Peter Mazsa, 19-Jul-2019.)

Assertion

Ref	Expression
inxpssres	|- ( R i^i ( A X. B ) ) C_ ( R |` A )

Proof of Theorem inxpssres

Step	Hyp	Ref	Expression
1		ssid 3624	. . . 4 |- A C_ A
2		ssv 3625	. . . 4 |- B C_ _V
3		xpss12 5225	. . . 4 |- ( ( A C_ A /\ B C_ _V ) -> ( A X. B ) C_ ( A X. _V ) )
4	1, 2, 3	mp2an 708	. . 3 |- ( A X. B ) C_ ( A X. _V )
5		sslin 3839	. . 3 |- ( ( A X. B ) C_ ( A X. _V ) -> ( R i^i ( A X. B ) ) C_ ( R i^i ( A X. _V ) ) )
6	4, 5	ax-mp 5	. 2 |- ( R i^i ( A X. B ) ) C_ ( R i^i ( A X. _V ) )
7		df-res 5126	. 2 |- ( R |` A ) = ( R i^i ( A X. _V ) )
8	6, 7	sseqtr4i 3638	1 |- ( R i^i ( A X. B ) ) C_ ( R |` A )

Syntax hints:   _Vcvv 3200   i^i cin 3573   C_ wss 3574   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-v 3202  df-in 3581  df-ss 3588  df-opab 4713  df-xp 5120  df-res 5126

This theorem is referenced by:  idreseqidinxp  34080  idinxpres  34088