MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  opelopab2a Structured version   Unicode version

Theorem opelopab2a 4616
Description: Ordered pair membership in an ordered pair class abstraction. (Contributed by Mario Carneiro, 19-Dec-2013.)
Hypothesis
Ref Expression
opelopabga.1  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
opelopab2a  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ( ( x  e.  C  /\  y  e.  D
)  /\  ph ) }  <->  ps ) )
Distinct variable groups:    x, y, A    x, B, y    ps, x, y    x, C, y   
x, D, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem opelopab2a
StepHypRef Expression
1 eleq1 2503 . . . . 5  |-  ( x  =  A  ->  (
x  e.  C  <->  A  e.  C ) )
2 eleq1 2503 . . . . 5  |-  ( y  =  B  ->  (
y  e.  D  <->  B  e.  D ) )
31, 2bi2anan9 868 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( x  e.  C  /\  y  e.  D )  <->  ( A  e.  C  /\  B  e.  D ) ) )
4 opelopabga.1 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
53, 4anbi12d 710 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( ( x  e.  C  /\  y  e.  D )  /\  ph ) 
<->  ( ( A  e.  C  /\  B  e.  D )  /\  ps ) ) )
65opelopabga 4614 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ( ( x  e.  C  /\  y  e.  D
)  /\  ph ) }  <-> 
( ( A  e.  C  /\  B  e.  D )  /\  ps ) ) )
76bianabs 875 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ( ( x  e.  C  /\  y  e.  D
)  /\  ph ) }  <->  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   <.cop 3895   {copab 4361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pr 4543
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-rab 2736  df-v 2986  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-opab 4363
This theorem is referenced by:  opelopab2  4621  brab2a  4900  brab2ga  4924  prdsleval  14427  isperp  23115
  Copyright terms: Public domain W3C validator