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Theorem opelopab3 26420
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 5003 . . . . . . 7  |-  Rel  { <. x ,  y >.  |  ph }
2 df-rel 4887 . . . . . . 7  |-  ( Rel 
{ <. x ,  y
>.  |  ph }  <->  { <. x ,  y >.  |  ph }  C_  ( _V  X.  _V ) )
31, 2mpbi 201 . . . . . 6  |-  { <. x ,  y >.  |  ph }  C_  ( _V  X.  _V )
43sseli 3346 . . . . 5  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  ->  <. A ,  B >.  e.  ( _V  X.  _V ) )
5 opelxp1 4913 . . . . 5  |-  ( <. A ,  B >.  e.  ( _V  X.  _V )  ->  A  e.  _V )
64, 5syl 16 . . . 4  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  ->  A  e.  _V )
76anim1i 553 . . 3  |-  ( (
<. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  /\  B  e.  D )  ->  ( A  e.  _V  /\  B  e.  D ) )
87ancoms 441 . 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 2966 . . . . 5  |-  ( A  e.  C  ->  A  e.  _V )
119, 10syl 16 . . . 4  |-  ( ch 
->  A  e.  _V )
1211anim1i 553 . . 3  |-  ( ( ch  /\  B  e.  D )  ->  ( A  e.  _V  /\  B  e.  D ) )
1312ancoms 441 . 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 4475 . 2  |-  ( ( A  e.  _V  /\  B  e.  D )  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph } 
<->  ch ) )
178, 13, 16pm5.21nd 870 1  |-  ( B  e.  D  ->  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  <->  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958    C_ wss 3322   <.cop 3819   {copab 4267    X. cxp 4878   Rel wrel 4885
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-opab 4269  df-xp 4886  df-rel 4887
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