Theorem idinxpssinxp2 34089

Description: Identity intersection with a square Cartesian product in subclass relation with an intersection with the same Cartesian product. (Contributed by Peter Mazsa, 4-Mar-2019.)

Assertion

Ref	Expression
idinxpssinxp2	|- ( ( _I i^i ( A X. A ) ) C_ ( R i^i ( A X. A ) ) <-> A. x e. A x R x )

Distinct variable groups:   x, A   x, R

Proof of Theorem idinxpssinxp2

Step	Hyp	Ref	Expression
1		idinxpres 34088	. . . 4 |- ( _I i^i ( A X. A ) ) = ( _I |` A )
2	1	sseq1i 3629	. . 3 |- ( ( _I i^i ( A X. A ) ) C_ ( R i^i ( A X. A ) ) <-> ( _I |` A ) C_ ( R i^i ( A X. A ) ) )
3		issref 5509	. . 3 |- ( ( _I |` A ) C_ ( R i^i ( A X. A ) ) <-> A. x e. A x ( R i^i ( A X. A ) ) x )
4		brinxp2ALTV 34034	. . . . 5 |- ( x ( R i^i ( A X. A ) ) x <-> ( ( x e. A /\ x e. A ) /\ x R x ) )
5		pm4.24 675	. . . . . 6 |- ( x e. A <-> ( x e. A /\ x e. A ) )
6	5	anbi1i 731	. . . . 5 |- ( ( x e. A /\ x R x ) <-> ( ( x e. A /\ x e. A ) /\ x R x ) )
7	4, 6	bitr4i 267	. . . 4 |- ( x ( R i^i ( A X. A ) ) x <-> ( x e. A /\ x R x ) )
8	7	ralbii 2980	. . 3 |- ( A. x e. A x ( R i^i ( A X. A ) ) x <-> A. x e. A ( x e. A /\ x R x ) )
9	2, 3, 8	3bitri 286	. 2 |- ( ( _I i^i ( A X. A ) ) C_ ( R i^i ( A X. A ) ) <-> A. x e. A ( x e. A /\ x R x ) )
10		ralanid 34010	. 2 |- ( A. x e. A ( x e. A /\ x R x ) <-> A. x e. A x R x )
11	9, 10	bitri 264	1 |- ( ( _I i^i ( A X. A ) ) C_ ( R i^i ( A X. A ) ) <-> A. x e. A x R x )

Syntax hints:   <-> wb 196   /\ wa 384   e. wcel 1990   A.wral 2912   i^i cin 3573   C_ wss 3574   class class class wbr 4653   _I cid 5023   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  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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-fun 5890  df-fn 5891

This theorem is referenced by:  idinxpssinxp3  34090  idinxpssinxp4  34091