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Theorem setlikespec 25213
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 2904 . . . . . 6  |-  x  e. 
_V
21elpred 25203 . . . . 5  |-  ( X  e.  A  ->  (
x  e.  Pred ( R ,  A ,  X )  <->  ( x  e.  A  /\  x R X ) ) )
32adantr 452 . . . 4  |-  ( ( X  e.  A  /\  R Se  A )  ->  (
x  e.  Pred ( R ,  A ,  X )  <->  ( x  e.  A  /\  x R X ) ) )
43abbi2dv 2504 . . 3  |-  ( ( X  e.  A  /\  R Se  A )  ->  Pred ( R ,  A ,  X )  =  {
x  |  ( x  e.  A  /\  x R X ) } )
5 df-rab 2660 . . 3  |-  { x  e.  A  |  x R X }  =  {
x  |  ( x  e.  A  /\  x R X ) }
64, 5syl6reqr 2440 . 2  |-  ( ( X  e.  A  /\  R Se  A )  ->  { x  e.  A  |  x R X }  =  Pred ( R ,  A ,  X ) )
7 seex 4488 . . 3  |-  ( ( R Se  A  /\  X  e.  A )  ->  { x  e.  A  |  x R X }  e.  _V )
87ancoms 440 . 2  |-  ( ( X  e.  A  /\  R Se  A )  ->  { x  e.  A  |  x R X }  e.  _V )
96, 8eqeltrrd 2464 1  |-  ( ( X  e.  A  /\  R Se  A )  ->  Pred ( R ,  A ,  X )  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1717   {cab 2375   {crab 2655   _Vcvv 2901   class class class wbr 4155   Se wse 4482   Predcpred 25193
This theorem is referenced by:  trpredtr  25259  trpredmintr  25260  trpredelss  25261  dftrpred3g  25262  trpredpo  25264  trpredrec  25267  frmin  25268  wfrlem15  25296
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 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346
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-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-br 4156  df-opab 4210  df-se 4485  df-xp 4826  df-cnv 4828  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-pred 25194
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