Theorem inxpss 34082

Description: Two ways to say that an intersection with a Cartesian product is a subclass. (Contributed by Peter Mazsa, 16-Jul-2019.)

Assertion

Ref	Expression
inxpss	|- ( ( R i^i ( A X. B ) ) C_ S <-> A. x e. A A. y e. B ( x R y -> x S y ) )

Distinct variable groups:   x, A, y   x, B, y   x, R, y   x, S, y

Proof of Theorem inxpss

Step	Hyp	Ref	Expression
1		brinxp2ALTV 34034	. . . . 5 |- ( x ( R i^i ( A X. B ) ) y <-> ( ( x e. A /\ y e. B ) /\ x R y ) )
2	1	imbi1i 339	. . . 4 |- ( ( x ( R i^i ( A X. B ) ) y -> x S y ) <-> ( ( ( x e. A /\ y e. B ) /\ x R y ) -> x S y ) )
3		impexp 462	. . . 4 |- ( ( ( ( x e. A /\ y e. B ) /\ x R y ) -> x S y ) <-> ( ( x e. A /\ y e. B ) -> ( x R y -> x S y ) ) )
4	2, 3	bitri 264	. . 3 |- ( ( x ( R i^i ( A X. B ) ) y -> x S y ) <-> ( ( x e. A /\ y e. B ) -> ( x R y -> x S y ) ) )
5	4	2albii 1748	. 2 |- ( A. x A. y ( x ( R i^i ( A X. B ) ) y -> x S y ) <-> A. x A. y ( ( x e. A /\ y e. B ) -> ( x R y -> x S y ) ) )
6		relinxp 34069	. . 3 |- Rel ( R i^i ( A X. B ) )
7		ssrel3 34067	. . 3 |- ( Rel ( R i^i ( A X. B ) ) -> ( ( R i^i ( A X. B ) ) C_ S <-> A. x A. y ( x ( R i^i ( A X. B ) ) y -> x S y ) ) )
8	6, 7	ax-mp 5	. 2 |- ( ( R i^i ( A X. B ) ) C_ S <-> A. x A. y ( x ( R i^i ( A X. B ) ) y -> x S y ) )
9		r2al 2939	. 2 |- ( A. x e. A A. y e. B ( x R y -> x S y ) <-> A. x A. y ( ( x e. A /\ y e. B ) -> ( x R y -> x S y ) ) )
10	5, 8, 9	3bitr4i 292	1 |- ( ( R i^i ( A X. B ) ) C_ S <-> A. x e. A A. y e. B ( x R y -> x S y ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   e. wcel 1990   A.wral 2912   i^i cin 3573   C_ wss 3574   class class class wbr 4653   X. cxp 5112   Rel wrel 5119

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  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  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-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121

This theorem is referenced by:  idinxpss  34083