Theorem seex 4336

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 4334	. 2 |- ( R Se A <-> A. y e. A { x e. A | x R y } e. _V )
2		breq2 4008	. . . . 5 |- ( y = B -> ( x R y <-> x R B ) )
3	2	rabbidv 2727	. . . 4 |- ( y = B -> { x e. A | x R y } = { x e. A | x R B } )
4	3	eleq1d 2246	. . 3 |- ( y = B -> ( { x e. A | x R y } e. _V <-> { x e. A | x R B } e. _V ) )
5	4	rspccva 2841	. 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 1353   e. wcel 2148   A.wral 2455   {crab 2459   _Vcvv 2738   class class class wbr 4004   Se wse 4330

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rab 2464  df-v 2740  df-un 3134  df-sn 3599  df-pr 3600  df-op 3602  df-br 4005  df-se 4334

This theorem is referenced by:  sefvex  5537