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Theorem setlikess 25409
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 3367 . . 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 25386 . . . . 5  |-  ( A 
C_  B  ->  Pred ( R ,  A ,  x )  C_  Pred ( R ,  B ,  x ) )
3 ssexg 4309 . . . . . 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 2745 . . 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 1721   A.wral 2666   _Vcvv 2916    C_ wss 3280   Predcpred 25381
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-v 2918  df-in 3287  df-ss 3294  df-pred 25382
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