Theorem seex 4265

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 4263	. 2 |- ( R Se A <-> A. y e. A { x e. A | x R y } e. _V )
2		breq2 3941	. . . . 5 |- ( y = B -> ( x R y <-> x R B ) )
3	2	rabbidv 2678	. . . 4 |- ( y = B -> { x e. A | x R y } = { x e. A | x R B } )
4	3	eleq1d 2209	. . 3 |- ( y = B -> ( { x e. A | x R y } e. _V <-> { x e. A | x R B } e. _V ) )
5	4	rspccva 2792	. 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 282	1 |- ( ( R Se A /\ B e. A ) -> { x e. A | x R B } e. _V )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1332   e. wcel 1481   A.wral 2417   {crab 2421   _Vcvv 2689   class class class wbr 3937   Se wse 4259

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rab 2426  df-v 2691  df-un 3080  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-se 4263

This theorem is referenced by:  sefvex  5450