Theorem seinxp 4674

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 4671	. . . . . 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 2713	. . . 4 |- ( x e. A -> { y e. A | y R x } = { y e. A | y ( R i^i ( A X. A ) ) x } )
4	3	eleq1d 2234	. . 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 2479	. 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 4310	. 2 |- ( R Se A <-> A. x e. A { y e. A | y R x } e. _V )
7		df-se 4310	. 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 2136   A.wral 2443   {crab 2447   _Vcvv 2725   i^i cin 3114   class class class wbr 3981   Se wse 4306   X. cxp 4601

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4099  ax-pow 4152  ax-pr 4186

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-ral 2448  df-rex 2449  df-rab 2452  df-v 2727  df-un 3119  df-in 3121  df-ss 3128  df-pw 3560  df-sn 3581  df-pr 3582  df-op 3584  df-br 3982  df-opab 4043  df-se 4310  df-xp 4609

This theorem is referenced by: (None)