Theorem seex 4370

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 4368	. 2 |- ( R Se A <-> A. y e. A { x e. A | x R y } e. _V )
2		breq2 4037	. . . . 5 |- ( y = B -> ( x R y <-> x R B ) )
3	2	rabbidv 2752	. . . 4 |- ( y = B -> { x e. A | x R y } = { x e. A | x R B } )
4	3	eleq1d 2265	. . 3 |- ( y = B -> ( { x e. A | x R y } e. _V <-> { x e. A | x R B } e. _V ) )
5	4	rspccva 2867	. 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 1364   e. wcel 2167   A.wral 2475   {crab 2479   _Vcvv 2763   class class class wbr 4033   Se wse 4364

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rab 2484  df-v 2765  df-un 3161  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-se 4368

This theorem is referenced by:  sefvex  5579