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Theorem setlikespec 24187
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 2791 . . . . . 6  |-  x  e. 
_V
21elpred 24177 . . . . 5  |-  ( X  e.  A  ->  (
x  e.  Pred ( R ,  A ,  X )  <->  ( x  e.  A  /\  x R X ) ) )
32adantr 451 . . . 4  |-  ( ( X  e.  A  /\  R Se  A )  ->  (
x  e.  Pred ( R ,  A ,  X )  <->  ( x  e.  A  /\  x R X ) ) )
43abbi2dv 2398 . . 3  |-  ( ( X  e.  A  /\  R Se  A )  ->  Pred ( R ,  A ,  X )  =  {
x  |  ( x  e.  A  /\  x R X ) } )
5 df-rab 2552 . . 3  |-  { x  e.  A  |  x R X }  =  {
x  |  ( x  e.  A  /\  x R X ) }
64, 5syl6reqr 2334 . 2  |-  ( ( X  e.  A  /\  R Se  A )  ->  { x  e.  A  |  x R X }  =  Pred ( R ,  A ,  X ) )
7 seex 4356 . . 3  |-  ( ( R Se  A  /\  X  e.  A )  ->  { x  e.  A  |  x R X }  e.  _V )
87ancoms 439 . 2  |-  ( ( X  e.  A  /\  R Se  A )  ->  { x  e.  A  |  x R X }  e.  _V )
96, 8eqeltrrd 2358 1  |-  ( ( X  e.  A  /\  R Se  A )  ->  Pred ( R ,  A ,  X )  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1684   {cab 2269   {crab 2547   _Vcvv 2788   class class class wbr 4023   Se wse 4350   Predcpred 24167
This theorem is referenced by:  trpredtr  24233  trpredmintr  24234  trpredelss  24235  dftrpred3g  24236  trpredpo  24238  trpredrec  24241  frmin  24242  wfrlem15  24270
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-se 4353  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-pred 24168
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