Theorem seinxp 4715

Description: Intersection of set-like relation with cross product of its field. (Contributed by Mario Carneiro, 22-Jun-2015.)

Assertion

Ref	Expression
seinxp	|- ( R Se A <-> ( R i^i ( A X. A ) ) Se A )

Proof of Theorem seinxp

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brinxp 4712	. . . . . 6 |- ( ( y e. A /\ x e. A ) -> ( y R x <-> y ( R i^i ( A X. A ) ) x ) )
2	1	ancoms 268	. . . . 5 |- ( ( x e. A /\ y e. A ) -> ( y R x <-> y ( R i^i ( A X. A ) ) x ) )
3	2	rabbidva 2740	. . . 4 |- ( x e. A -> { y e. A | y R x } = { y e. A | y ( R i^i ( A X. A ) ) x } )
4	3	eleq1d 2258	. . 3 |- ( x e. A -> ( { y e. A | y R x } e. _V <-> { y e. A | y ( R i^i ( A X. A ) ) x } e. _V ) )
5	4	ralbiia 2504	. 2 |- ( A. x e. A { y e. A | y R x } e. _V <-> A. x e. A { y e. A | y ( R i^i ( A X. A ) ) x } e. _V )
6		df-se 4351	. 2 |- ( R Se A <-> A. x e. A { y e. A | y R x } e. _V )
7		df-se 4351	. 2 |- ( ( R i^i ( A X. A ) ) Se A <-> A. x e. A { y e. A | y ( R i^i ( A X. A ) ) x } e. _V )
8	5, 6, 7	3bitr4i 212	1 |- ( R Se A <-> ( R i^i ( A X. A ) ) Se A )

Syntax hints:   <-> wb 105   e. wcel 2160   A.wral 2468   {crab 2472   _Vcvv 2752   i^i cin 3143   class class class wbr 4018   Se wse 4347   X. cxp 4642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-se 4351  df-xp 4650

This theorem is referenced by: (None)