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Theorem opelopab3 32107
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 elopaelxp 4912 . . . . 5  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  ->  <. A ,  B >.  e.  ( _V  X.  _V ) )
2 opelxp1 4872 . . . . 5  |-  ( <. A ,  B >.  e.  ( _V  X.  _V )  ->  A  e.  _V )
31, 2syl 17 . . . 4  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  ->  A  e.  _V )
43anim1i 578 . . 3  |-  ( (
<. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  /\  B  e.  D )  ->  ( A  e.  _V  /\  B  e.  D ) )
54ancoms 460 . 2  |-  ( ( B  e.  D  /\  <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph } )  ->  ( A  e. 
_V  /\  B  e.  D ) )
6 opelopab3.3 . . . . 5  |-  ( ch 
->  A  e.  C
)
7 elex 3040 . . . . 5  |-  ( A  e.  C  ->  A  e.  _V )
86, 7syl 17 . . . 4  |-  ( ch 
->  A  e.  _V )
98anim1i 578 . . 3  |-  ( ( ch  /\  B  e.  D )  ->  ( A  e.  _V  /\  B  e.  D ) )
109ancoms 460 . 2  |-  ( ( B  e.  D  /\  ch )  ->  ( A  e.  _V  /\  B  e.  D ) )
11 opelopab3.1 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
12 opelopab3.2 . . 3  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
1311, 12opelopabg 4719 . 2  |-  ( ( A  e.  _V  /\  B  e.  D )  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph } 
<->  ch ) )
145, 10, 13pm5.21nd 916 1  |-  ( B  e.  D  ->  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  <->  ch )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   _Vcvv 3031   <.cop 3965   {copab 4453    X. cxp 4837
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-opab 4455  df-xp 4845
This theorem is referenced by: (None)
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