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Theorem opelopab3 26373
Description: Ordered pair membership in an ordered pair class abstraction, with a reduced hypothesis. (Contributed by Jeff Madsen, 29-May-2011.)
Hypotheses
Ref Expression
opelopab3.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
opelopab3.2  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
opelopab3.3  |-  ( ch 
->  A  e.  C
)
Assertion
Ref Expression
opelopab3  |-  ( B  e.  D  ->  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  <->  ch )
)
Distinct variable groups:    x, A, y    x, B, y    ch, x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)    C( x, y)    D( x, y)

Proof of Theorem opelopab3
StepHypRef Expression
1 relopab 4812 . . . . . . 7  |-  Rel  { <. x ,  y >.  |  ph }
2 df-rel 4696 . . . . . . 7  |-  ( Rel 
{ <. x ,  y
>.  |  ph }  <->  { <. x ,  y >.  |  ph }  C_  ( _V  X.  _V ) )
31, 2mpbi 199 . . . . . 6  |-  { <. x ,  y >.  |  ph }  C_  ( _V  X.  _V )
43sseli 3176 . . . . 5  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  ->  <. A ,  B >.  e.  ( _V  X.  _V ) )
5 opelxp1 4722 . . . . 5  |-  ( <. A ,  B >.  e.  ( _V  X.  _V )  ->  A  e.  _V )
64, 5syl 15 . . . 4  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  ->  A  e.  _V )
76anim1i 551 . . 3  |-  ( (
<. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  /\  B  e.  D )  ->  ( A  e.  _V  /\  B  e.  D ) )
87ancoms 439 . 2  |-  ( ( B  e.  D  /\  <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph } )  ->  ( A  e. 
_V  /\  B  e.  D ) )
9 opelopab3.3 . . . . 5  |-  ( ch 
->  A  e.  C
)
10 elex 2796 . . . . 5  |-  ( A  e.  C  ->  A  e.  _V )
119, 10syl 15 . . . 4  |-  ( ch 
->  A  e.  _V )
1211anim1i 551 . . 3  |-  ( ( ch  /\  B  e.  D )  ->  ( A  e.  _V  /\  B  e.  D ) )
1312ancoms 439 . 2  |-  ( ( B  e.  D  /\  ch )  ->  ( A  e.  _V  /\  B  e.  D ) )
14 opelopab3.1 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
15 opelopab3.2 . . 3  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
1614, 15opelopabg 4283 . 2  |-  ( ( A  e.  _V  /\  B  e.  D )  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph } 
<->  ch ) )
178, 13, 16pm5.21nd 868 1  |-  ( B  e.  D  ->  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  <->  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   <.cop 3643   {copab 4076    X. cxp 4687   Rel wrel 4694
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-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  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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