Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  opelopab3 Structured version   Unicode version

Theorem opelopab3 28759
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 5075 . . . . . . 7  |-  Rel  { <. x ,  y >.  |  ph }
2 df-rel 4956 . . . . . . 7  |-  ( Rel 
{ <. x ,  y
>.  |  ph }  <->  { <. x ,  y >.  |  ph }  C_  ( _V  X.  _V ) )
31, 2mpbi 208 . . . . . 6  |-  { <. x ,  y >.  |  ph }  C_  ( _V  X.  _V )
43sseli 3461 . . . . 5  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  ->  <. A ,  B >.  e.  ( _V  X.  _V ) )
5 opelxp1 4981 . . . . 5  |-  ( <. A ,  B >.  e.  ( _V  X.  _V )  ->  A  e.  _V )
64, 5syl 16 . . . 4  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  ->  A  e.  _V )
76anim1i 568 . . 3  |-  ( (
<. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  /\  B  e.  D )  ->  ( A  e.  _V  /\  B  e.  D ) )
87ancoms 453 . 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 3087 . . . . 5  |-  ( A  e.  C  ->  A  e.  _V )
119, 10syl 16 . . . 4  |-  ( ch 
->  A  e.  _V )
1211anim1i 568 . . 3  |-  ( ( ch  /\  B  e.  D )  ->  ( A  e.  _V  /\  B  e.  D ) )
1312ancoms 453 . 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 4716 . 2  |-  ( ( A  e.  _V  /\  B  e.  D )  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph } 
<->  ch ) )
178, 13, 16pm5.21nd 893 1  |-  ( B  e.  D  ->  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  <->  ch )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3078    C_ wss 3437   <.cop 3992   {copab 4458    X. cxp 4947   Rel wrel 4954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-opab 4460  df-xp 4955  df-rel 4956
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator