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Theorem seex 4257
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.
StepHypRef Expression
1 df-se 4255 . 2  |-  ( R Se  A  <->  A. y  e.  A  { x  e.  A  |  x R y }  e.  _V )
2 breq2 3933 . . . . 5  |-  ( y  =  B  ->  (
x R y  <->  x R B ) )
32rabbidv 2675 . . . 4  |-  ( y  =  B  ->  { x  e.  A  |  x R y }  =  { x  e.  A  |  x R B }
)
43eleq1d 2208 . . 3  |-  ( y  =  B  ->  ( { x  e.  A  |  x R y }  e.  _V  <->  { x  e.  A  |  x R B }  e.  _V ) )
54rspccva 2788 . 2  |-  ( ( A. y  e.  A  { x  e.  A  |  x R y }  e.  _V  /\  B  e.  A )  ->  { x  e.  A  |  x R B }  e.  _V )
61, 5sylanb 282 1  |-  ( ( R Se  A  /\  B  e.  A )  ->  { x  e.  A  |  x R B }  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480   A.wral 2416   {crab 2420   _Vcvv 2686   class class class wbr 3929   Se wse 4251
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rab 2425  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-se 4255
This theorem is referenced by:  sefvex  5442
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