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Theorem seex 4548
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 4545 . 2  |-  ( R Se  A  <->  A. y  e.  A  { x  e.  A  |  x R y }  e.  _V )
2 breq2 4219 . . . . 5  |-  ( y  =  B  ->  (
x R y  <->  x R B ) )
32rabbidv 2950 . . . 4  |-  ( y  =  B  ->  { x  e.  A  |  x R y }  =  { x  e.  A  |  x R B }
)
43eleq1d 2504 . . 3  |-  ( y  =  B  ->  ( { x  e.  A  |  x R y }  e.  _V  <->  { x  e.  A  |  x R B }  e.  _V ) )
54rspccva 3053 . 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 460 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 360    = wceq 1653    e. wcel 1726   A.wral 2707   {crab 2711   _Vcvv 2958   class class class wbr 4215   Se wse 4542
This theorem is referenced by:  wereu2  4582  fnse  6466  ordtypelem10  7499  setlikespec  25467
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-se 4545
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