Theorem seinxp 4682

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 4679	. . . . . 6 |- ( ( y e. A /\ x e. A ) -> ( y R x <-> y ( R i^i ( A X. A ) ) x ) )
2	1	ancoms 266	. . . . 5 |- ( ( x e. A /\ y e. A ) -> ( y R x <-> y ( R i^i ( A X. A ) ) x ) )
3	2	rabbidva 2718	. . . 4 |- ( x e. A -> { y e. A | y R x } = { y e. A | y ( R i^i ( A X. A ) ) x } )
4	3	eleq1d 2239	. . 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 2484	. 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 4318	. 2 |- ( R Se A <-> A. x e. A { y e. A | y R x } e. _V )
7		df-se 4318	. 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 211	1 |- ( R Se A <-> ( R i^i ( A X. A ) ) Se A )

Syntax hints:   <-> wb 104   e. wcel 2141   A.wral 2448   {crab 2452   _Vcvv 2730   i^i cin 3120   class class class wbr 3989   Se wse 4314   X. cxp 4609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-se 4318  df-xp 4617

This theorem is referenced by: (None)