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Theorem relssdv 4779
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 1618 . 2  |-  ( ph  ->  A. x A. y
( <. x ,  y
>.  e.  A  ->  <. x ,  y >.  e.  B
) )
3 relssdv.1 . . 3  |-  ( ph  ->  Rel  A )
4 ssrel 4776 . . 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 1527    e. wcel 1684    C_ wss 3152   <.cop 3643   Rel wrel 4694
This theorem is referenced by:  relssres  4992  poirr2  5067  sofld  5121  relssdmrn  5193  funcres2  13772  wunfunc  13773  fthres2  13806  pospo  14107  subrgdvds  15559  opsrtoslem2  16226  txcls  17299  txdis1cn  17329  txkgen  17346  divstgplem  17803  ovoliunlem1  18861  dvres2  19262  cvmlift2lem12  23845  dib2dim  31433  dih2dimbALTN  31435  dihmeetlem1N  31480  dihglblem5apreN  31481  dihmeetlem13N  31509  dihjatcclem4  31611
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-opab 4078  df-xp 4695  df-rel 4696
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