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Theorem seex 4438
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 4435 . 2  |-  ( R Se  A  <->  A. y  e.  A  { x  e.  A  |  x R y }  e.  _V )
2 breq2 4108 . . . . 5  |-  ( y  =  B  ->  (
x R y  <->  x R B ) )
32rabbidv 2856 . . . 4  |-  ( y  =  B  ->  { x  e.  A  |  x R y }  =  { x  e.  A  |  x R B }
)
43eleq1d 2424 . . 3  |-  ( y  =  B  ->  ( { x  e.  A  |  x R y }  e.  _V  <->  { x  e.  A  |  x R B }  e.  _V ) )
54rspccva 2959 . 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 458 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 358    = wceq 1642    e. wcel 1710   A.wral 2619   {crab 2623   _Vcvv 2864   class class class wbr 4104   Se wse 4432
This theorem is referenced by:  wereu2  4472  fnse  6319  ordtypelem10  7332  setlikespec  24745
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ral 2624  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-br 4105  df-se 4435
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