Theorem seex 4458

Description: The R-preimage of an element of the base set in a set-like relation is a set. (Contributed by Mario Carneiro, 19-Nov-2014.)

Assertion

Ref	Expression
seex	|- ( ( R Se A /\ B e. A ) -> { x e. A | x R B } e. _V )

Distinct variable groups:   x, A   x, B   x, R

Proof of Theorem seex

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-se 4456	. 2 |- ( R Se A <-> A. y e. A { x e. A | x R y } e. _V )
2		breq2 4115	. . . . 5 |- ( y = B -> ( x R y <-> x R B ) )
3	2	rabbidv 2804	. . . 4 |- ( y = B -> { x e. A | x R y } = { x e. A | x R B } )
4	3	eleq1d 2303	. . 3 |- ( y = B -> ( { x e. A | x R y } e. _V <-> { x e. A | x R B } e. _V ) )
5	4	rspccva 2922	. 2 |- ( ( A. y e. A { x e. A | x R y } e. _V /\ B e. A ) -> { x e. A | x R B } e. _V )
6	1, 5	sylanb 284	1 |- ( ( R Se A /\ B e. A ) -> { x e. A | x R B } e. _V )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   A.wral 2522   {crab 2526   _Vcvv 2815   class class class wbr 4111   Se wse 4452

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rab 2531  df-v 2817  df-un 3217  df-sn 3697  df-pr 3698  df-op 3700  df-br 4112  df-se 4456

This theorem is referenced by:  sefvex  5693