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Theorem setlikespec 25447
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 2951 . . . . . 6  |-  x  e. 
_V
21elpred 25437 . . . . 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 2550 . . 3  |-  ( ( X  e.  A  /\  R Se  A )  ->  Pred ( R ,  A ,  X )  =  {
x  |  ( x  e.  A  /\  x R X ) } )
5 df-rab 2706 . . 3  |-  { x  e.  A  |  x R X }  =  {
x  |  ( x  e.  A  /\  x R X ) }
64, 5syl6reqr 2486 . 2  |-  ( ( X  e.  A  /\  R Se  A )  ->  { x  e.  A  |  x R X }  =  Pred ( R ,  A ,  X ) )
7 seex 4537 . . 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 2510 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 1725   {cab 2421   {crab 2701   _Vcvv 2948   class class class wbr 4204   Se wse 4531   Predcpred 25426
This theorem is referenced by:  trpredtr  25493  trpredmintr  25494  trpredelss  25495  dftrpred3g  25496  trpredpo  25498  trpredrec  25501  frmin  25502  wfrlem15  25537
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-se 4534  df-xp 4876  df-cnv 4878  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-pred 25427
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