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Theorem opelopab3 25522
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 4849 . . . . . . 7  |-  Rel  { <. x ,  y >.  |  ph }
2 df-rel 4733 . . . . . . 7  |-  ( Rel 
{ <. x ,  y
>.  |  ph }  <->  { <. x ,  y >.  |  ph }  C_  ( _V  X.  _V ) )
31, 2mpbi 199 . . . . . 6  |-  { <. x ,  y >.  |  ph }  C_  ( _V  X.  _V )
43sseli 3210 . . . . 5  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  ->  <. A ,  B >.  e.  ( _V  X.  _V ) )
5 opelxp1 4759 . . . . 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 2830 . . . . 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 4320 . 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 1633    e. wcel 1701   _Vcvv 2822    C_ wss 3186   <.cop 3677   {copab 4113    X. cxp 4724   Rel wrel 4731
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-opab 4115  df-xp 4732  df-rel 4733
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