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Theorem seex 4513
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 4510 . 2  |-  ( R Se  A  <->  A. y  e.  A  { x  e.  A  |  x R y }  e.  _V )
2 breq2 4184 . . . . 5  |-  ( y  =  B  ->  (
x R y  <->  x R B ) )
32rabbidv 2916 . . . 4  |-  ( y  =  B  ->  { x  e.  A  |  x R y }  =  { x  e.  A  |  x R B }
)
43eleq1d 2478 . . 3  |-  ( y  =  B  ->  ( { x  e.  A  |  x R y }  e.  _V  <->  { x  e.  A  |  x R B }  e.  _V ) )
54rspccva 3019 . 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 459 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 359    = wceq 1649    e. wcel 1721   A.wral 2674   {crab 2678   _Vcvv 2924   class class class wbr 4180   Se wse 4507
This theorem is referenced by:  wereu2  4547  fnse  6430  ordtypelem10  7460  setlikespec  25409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ral 2679  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-br 4181  df-se 4510
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