Theorem idinxpssinxp 34087

Description: Two ways to say that intersections with Cartesian products are in a subclass relation, special case of inxpss2 34085. (Contributed by Peter Mazsa, 6-Mar-2019.)

Assertion

Ref	Expression
idinxpssinxp	|- ( ( _I i^i ( A X. B ) ) C_ ( R i^i ( A X. B ) ) <-> A. x e. A A. y e. B ( x = y -> x R y ) )

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

Proof of Theorem idinxpssinxp

Step	Hyp	Ref	Expression
1		inxpss2 34085	. 2 |- ( ( _I i^i ( A X. B ) ) C_ ( R i^i ( A X. B ) ) <-> A. x e. A A. y e. B ( x _I y -> x R y ) )
2		ideqg 5273	. . . . 5 |- ( y e. _V -> ( x _I y <-> x = y ) )
3	2	elv 33983	. . . 4 |- ( x _I y <-> x = y )
4	3	imbi1i 339	. . 3 |- ( ( x _I y -> x R y ) <-> ( x = y -> x R y ) )
5	4	2ralbii 2981	. 2 |- ( A. x e. A A. y e. B ( x _I y -> x R y ) <-> A. x e. A A. y e. B ( x = y -> x R y ) )
6	1, 5	bitri 264	1 |- ( ( _I i^i ( A X. B ) ) C_ ( R i^i ( A X. B ) ) <-> A. x e. A A. y e. B ( x = y -> x R y ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   A.wral 2912   _Vcvv 3200   i^i cin 3573   C_ wss 3574   class class class wbr 4653   _I cid 5023   X. cxp 5112

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-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-id 5024  df-xp 5120  df-rel 5121

This theorem is referenced by:  idinxpssinxp4  34091