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Theorem relssdv 4795
Description: Deduction from subclass principle for relations. (Contributed by NM, 11-Sep-2004.)
Hypotheses
Ref Expression
relssdv.1  |-  ( ph  ->  Rel  A )
relssdv.2  |-  ( ph  ->  ( <. x ,  y
>.  e.  A  ->  <. x ,  y >.  e.  B
) )
Assertion
Ref Expression
relssdv  |-  ( ph  ->  A  C_  B )
Distinct variable groups:    x, y, A    x, B, y    ph, x, y

Proof of Theorem relssdv
StepHypRef Expression
1 relssdv.2 . . 3  |-  ( ph  ->  ( <. x ,  y
>.  e.  A  ->  <. x ,  y >.  e.  B
) )
21alrimivv 1622 . 2  |-  ( ph  ->  A. x A. y
( <. x ,  y
>.  e.  A  ->  <. x ,  y >.  e.  B
) )
3 relssdv.1 . . 3  |-  ( ph  ->  Rel  A )
4 ssrel 4792 . . 3  |-  ( Rel 
A  ->  ( A  C_  B  <->  A. x A. y
( <. x ,  y
>.  e.  A  ->  <. x ,  y >.  e.  B
) ) )
53, 4syl 15 . 2  |-  ( ph  ->  ( A  C_  B  <->  A. x A. y (
<. x ,  y >.  e.  A  ->  <. x ,  y >.  e.  B
) ) )
62, 5mpbird 223 1  |-  ( ph  ->  A  C_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1530    e. wcel 1696    C_ wss 3165   <.cop 3656   Rel wrel 4710
This theorem is referenced by:  relssres  5008  poirr2  5083  sofld  5137  relssdmrn  5209  funcres2  13788  wunfunc  13789  fthres2  13822  pospo  14123  subrgdvds  15575  opsrtoslem2  16242  txcls  17315  txdis1cn  17345  txkgen  17362  divstgplem  17819  ovoliunlem1  18877  dvres2  19278  cvmlift2lem12  23860  dib2dim  32055  dih2dimbALTN  32057  dihmeetlem1N  32102  dihglblem5apreN  32103  dihmeetlem13N  32131  dihjatcclem4  32233
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  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-ne 2461  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-opab 4094  df-xp 4711  df-rel 4712
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