Theorem seinxp 4789

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 4786	. . . . . 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 2787	. . . 4 |- ( x e. A -> { y e. A | y R x } = { y e. A | y ( R i^i ( A X. A ) ) x } )
4	3	eleq1d 2298	. . 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 2544	. 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 4423	. 2 |- ( R Se A <-> A. x e. A { y e. A | y R x } e. _V )
7		df-se 4423	. 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 2200   A.wral 2508   {crab 2512   _Vcvv 2799   i^i cin 3196   class class class wbr 4082   Se wse 4419   X. cxp 4716

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-se 4423  df-xp 4724

This theorem is referenced by: (None)