Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  setlikespec Structured version   Unicode version

Theorem setlikespec 27782
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 3071 . . . . . 6  |-  x  e. 
_V
21elpred 27772 . . . . 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 2588 . . 3  |-  ( ( X  e.  A  /\  R Se  A )  ->  Pred ( R ,  A ,  X )  =  {
x  |  ( x  e.  A  /\  x R X ) } )
5 df-rab 2804 . . 3  |-  { x  e.  A  |  x R X }  =  {
x  |  ( x  e.  A  /\  x R X ) }
64, 5syl6reqr 2511 . 2  |-  ( ( X  e.  A  /\  R Se  A )  ->  { x  e.  A  |  x R X }  =  Pred ( R ,  A ,  X ) )
7 seex 4781 . . 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 2540 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 1758   {cab 2436   {crab 2799   _Vcvv 3068   class class class wbr 4390   Se wse 4775   Predcpred 27758
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-sbc 3285  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-sn 3976  df-pr 3978  df-op 3982  df-br 4391  df-opab 4449  df-se 4778  df-xp 4944  df-cnv 4946  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-pred 27759
This theorem is referenced by:  trpredtr  27828  trpredmintr  27829  trpredelss  27830  dftrpred3g  27831  trpredpo  27833  trpredrec  27836  frmin  27837  wfrlem15  27872
  Copyright terms: Public domain W3C validator