Theorem seinxp 4730

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 4727	. . . . . 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 2748	. . . 4 |- ( x e. A -> { y e. A | y R x } = { y e. A | y ( R i^i ( A X. A ) ) x } )
4	3	eleq1d 2262	. . 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 2508	. 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 4364	. 2 |- ( R Se A <-> A. x e. A { y e. A | y R x } e. _V )
7		df-se 4364	. 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 2164   A.wral 2472   {crab 2476   _Vcvv 2760   i^i cin 3152   class class class wbr 4029   Se wse 4360   X. cxp 4657

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 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-se 4364  df-xp 4665

This theorem is referenced by: (None)