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Theorem setlikespec 28694
Description: If  R is set-like in  A, then all predecessors classes of elements of  A exist. (Contributed by Scott Fenton, 20-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Assertion
Ref Expression
setlikespec  |-  ( ( X  e.  A  /\  R Se  A )  ->  Pred ( R ,  A ,  X )  e.  _V )

Proof of Theorem setlikespec
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 vex 3109 . . . . . 6  |-  x  e. 
_V
21elpred 28684 . . . . 5  |-  ( X  e.  A  ->  (
x  e.  Pred ( R ,  A ,  X )  <->  ( x  e.  A  /\  x R X ) ) )
32adantr 465 . . . 4  |-  ( ( X  e.  A  /\  R Se  A )  ->  (
x  e.  Pred ( R ,  A ,  X )  <->  ( x  e.  A  /\  x R X ) ) )
43abbi2dv 2597 . . 3  |-  ( ( X  e.  A  /\  R Se  A )  ->  Pred ( R ,  A ,  X )  =  {
x  |  ( x  e.  A  /\  x R X ) } )
5 df-rab 2816 . . 3  |-  { x  e.  A  |  x R X }  =  {
x  |  ( x  e.  A  /\  x R X ) }
64, 5syl6reqr 2520 . 2  |-  ( ( X  e.  A  /\  R Se  A )  ->  { x  e.  A  |  x R X }  =  Pred ( R ,  A ,  X ) )
7 seex 4835 . . 3  |-  ( ( R Se  A  /\  X  e.  A )  ->  { x  e.  A  |  x R X }  e.  _V )
87ancoms 453 . 2  |-  ( ( X  e.  A  /\  R Se  A )  ->  { x  e.  A  |  x R X }  e.  _V )
96, 8eqeltrrd 2549 1  |-  ( ( X  e.  A  /\  R Se  A )  ->  Pred ( R ,  A ,  X )  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1762   {cab 2445   {crab 2811   _Vcvv 3106   class class class wbr 4440   Se wse 4829   Predcpred 28670
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-br 4441  df-opab 4499  df-se 4832  df-xp 4998  df-cnv 5000  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-pred 28671
This theorem is referenced by:  trpredtr  28740  trpredmintr  28741  trpredelss  28742  dftrpred3g  28743  trpredpo  28745  trpredrec  28748  frmin  28749  wfrlem15  28784
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