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Theorem setlikess 25470
Description: If  R is set-like over  A, then it is set-like over any subclass of  A. (Contributed by Scott Fenton, 28-Mar-2011.)
Assertion
Ref Expression
setlikess  |-  ( ( A  C_  B  /\  A. x  e.  B  Pred ( R ,  B ,  x )  e.  _V )  ->  A. x  e.  A  Pred ( R ,  A ,  x )  e.  _V )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    R( x)

Proof of Theorem setlikess
StepHypRef Expression
1 ssralv 3407 . . 3  |-  ( A 
C_  B  ->  ( A. x  e.  B  Pred ( R ,  B ,  x )  e.  _V  ->  A. x  e.  A  Pred ( R ,  B ,  x )  e.  _V ) )
2 predpredss 25445 . . . . 5  |-  ( A 
C_  B  ->  Pred ( R ,  A ,  x )  C_  Pred ( R ,  B ,  x ) )
3 ssexg 4349 . . . . . 6  |-  ( (
Pred ( R ,  A ,  x )  C_ 
Pred ( R ,  B ,  x )  /\  Pred ( R ,  B ,  x )  e.  _V )  ->  Pred ( R ,  A ,  x )  e.  _V )
43ex 424 . . . . 5  |-  ( Pred ( R ,  A ,  x )  C_  Pred ( R ,  B ,  x )  ->  ( Pred ( R ,  B ,  x )  e.  _V  ->  Pred ( R ,  A ,  x )  e.  _V ) )
52, 4syl 16 . . . 4  |-  ( A 
C_  B  ->  ( Pred ( R ,  B ,  x )  e.  _V  ->  Pred ( R ,  A ,  x )  e.  _V ) )
65ralimdv 2785 . . 3  |-  ( A 
C_  B  ->  ( A. x  e.  A  Pred ( R ,  B ,  x )  e.  _V  ->  A. x  e.  A  Pred ( R ,  A ,  x )  e.  _V ) )
71, 6syld 42 . 2  |-  ( A 
C_  B  ->  ( A. x  e.  B  Pred ( R ,  B ,  x )  e.  _V  ->  A. x  e.  A  Pred ( R ,  A ,  x )  e.  _V ) )
87imp 419 1  |-  ( ( A  C_  B  /\  A. x  e.  B  Pred ( R ,  B ,  x )  e.  _V )  ->  A. x  e.  A  Pred ( R ,  A ,  x )  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1725   A.wral 2705   _Vcvv 2956    C_ wss 3320   Predcpred 25438
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-v 2958  df-in 3327  df-ss 3334  df-pred 25439
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