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Theorem setlikess 24266
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 3250 . . 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 24243 . . . . 5  |-  ( A 
C_  B  ->  Pred ( R ,  A ,  x )  C_  Pred ( R ,  B ,  x ) )
3 ssexg 4176 . . . . . 6  |-  ( (
Pred ( R ,  A ,  x )  C_ 
Pred ( R ,  B ,  x )  /\  Pred ( R ,  B ,  x )  e.  _V )  ->  Pred ( R ,  A ,  x )  e.  _V )
43ex 423 . . . . 5  |-  ( Pred ( R ,  A ,  x )  C_  Pred ( R ,  B ,  x )  ->  ( Pred ( R ,  B ,  x )  e.  _V  ->  Pred ( R ,  A ,  x )  e.  _V ) )
52, 4syl 15 . . . 4  |-  ( A 
C_  B  ->  ( Pred ( R ,  B ,  x )  e.  _V  ->  Pred ( R ,  A ,  x )  e.  _V ) )
65ralimdv 2635 . . 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 40 . 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 418 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 358    e. wcel 1696   A.wral 2556   _Vcvv 2801    C_ wss 3165   Predcpred 24238
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-v 2803  df-in 3172  df-ss 3179  df-pred 24239
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