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Theorem opbrop 5027
Description: Ordered pair membership in a relation. Special case. (Contributed by NM, 5-Aug-1995.)
Hypotheses
Ref Expression
opbrop.1  |-  ( ( ( z  =  A  /\  w  =  B )  /\  ( v  =  C  /\  u  =  D ) )  -> 
( ph  <->  ps ) )
opbrop.2  |-  R  =  { <. x ,  y
>.  |  ( (
x  e.  ( S  X.  S )  /\  y  e.  ( S  X.  S ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ph )
) }
Assertion
Ref Expression
opbrop  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( <. A ,  B >. R <. C ,  D >.  <->  ps ) )
Distinct variable groups:    x, y,
z, w, v, u, A    x, B, y, z, w, v, u   
x, C, y, z, w, v, u    x, D, y, z, w, v, u    x, S, y, z, w, v, u    ph, x, y    ps, z, w, v, u
Allowed substitution hints:    ph( z, w, v, u)    ps( x, y)    R( x, y, z, w, v, u)

Proof of Theorem opbrop
StepHypRef Expression
1 opex 4667 . . . 4  |-  <. A ,  B >.  e.  _V
2 opex 4667 . . . 4  |-  <. C ,  D >.  e.  _V
3 eleq1 2526 . . . . . 6  |-  ( x  =  <. A ,  B >.  ->  ( x  e.  ( S  X.  S
)  <->  <. A ,  B >.  e.  ( S  X.  S ) ) )
43anbi1d 704 . . . . 5  |-  ( x  =  <. A ,  B >.  ->  ( ( x  e.  ( S  X.  S )  /\  y  e.  ( S  X.  S
) )  <->  ( <. A ,  B >.  e.  ( S  X.  S )  /\  y  e.  ( S  X.  S ) ) ) )
5 eqeq1 2458 . . . . . . . 8  |-  ( x  =  <. A ,  B >.  ->  ( x  = 
<. z ,  w >.  <->  <. A ,  B >.  =  <. z ,  w >. )
)
65anbi1d 704 . . . . . . 7  |-  ( x  =  <. A ,  B >.  ->  ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  <->  (
<. A ,  B >.  = 
<. z ,  w >.  /\  y  =  <. v ,  u >. ) ) )
76anbi1d 704 . . . . . 6  |-  ( x  =  <. A ,  B >.  ->  ( ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ph )  <->  ( ( <. A ,  B >.  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ph ) ) )
874exbidv 1685 . . . . 5  |-  ( x  =  <. A ,  B >.  ->  ( E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ph )  <->  E. z E. w E. v E. u ( ( <. A ,  B >.  = 
<. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ph ) ) )
94, 8anbi12d 710 . . . 4  |-  ( x  =  <. A ,  B >.  ->  ( ( ( x  e.  ( S  X.  S )  /\  y  e.  ( S  X.  S ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ph )
)  <->  ( ( <. A ,  B >.  e.  ( S  X.  S
)  /\  y  e.  ( S  X.  S
) )  /\  E. z E. w E. v E. u ( ( <. A ,  B >.  = 
<. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ph ) ) ) )
10 eleq1 2526 . . . . . 6  |-  ( y  =  <. C ,  D >.  ->  ( y  e.  ( S  X.  S
)  <->  <. C ,  D >.  e.  ( S  X.  S ) ) )
1110anbi2d 703 . . . . 5  |-  ( y  =  <. C ,  D >.  ->  ( ( <. A ,  B >.  e.  ( S  X.  S
)  /\  y  e.  ( S  X.  S
) )  <->  ( <. A ,  B >.  e.  ( S  X.  S )  /\  <. C ,  D >.  e.  ( S  X.  S ) ) ) )
12 eqeq1 2458 . . . . . . . 8  |-  ( y  =  <. C ,  D >.  ->  ( y  = 
<. v ,  u >.  <->  <. C ,  D >.  =  <. v ,  u >. )
)
1312anbi2d 703 . . . . . . 7  |-  ( y  =  <. C ,  D >.  ->  ( ( <. A ,  B >.  = 
<. z ,  w >.  /\  y  =  <. v ,  u >. )  <->  ( <. A ,  B >.  =  <. z ,  w >.  /\  <. C ,  D >.  =  <. v ,  u >. )
) )
1413anbi1d 704 . . . . . 6  |-  ( y  =  <. C ,  D >.  ->  ( ( (
<. A ,  B >.  = 
<. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ph ) 
<->  ( ( <. A ,  B >.  =  <. z ,  w >.  /\  <. C ,  D >.  =  <. v ,  u >. )  /\  ph ) ) )
15144exbidv 1685 . . . . 5  |-  ( y  =  <. C ,  D >.  ->  ( E. z E. w E. v E. u ( ( <. A ,  B >.  = 
<. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ph ) 
<->  E. z E. w E. v E. u ( ( <. A ,  B >.  =  <. z ,  w >.  /\  <. C ,  D >.  =  <. v ,  u >. )  /\  ph )
) )
1611, 15anbi12d 710 . . . 4  |-  ( y  =  <. C ,  D >.  ->  ( ( (
<. A ,  B >.  e.  ( S  X.  S
)  /\  y  e.  ( S  X.  S
) )  /\  E. z E. w E. v E. u ( ( <. A ,  B >.  = 
<. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ph ) )  <->  ( ( <. A ,  B >.  e.  ( S  X.  S
)  /\  <. C ,  D >.  e.  ( S  X.  S ) )  /\  E. z E. w E. v E. u ( ( <. A ,  B >.  = 
<. z ,  w >.  /\ 
<. C ,  D >.  = 
<. v ,  u >. )  /\  ph ) ) ) )
17 opbrop.2 . . . 4  |-  R  =  { <. x ,  y
>.  |  ( (
x  e.  ( S  X.  S )  /\  y  e.  ( S  X.  S ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ph )
) }
181, 2, 9, 16, 17brab 4722 . . 3  |-  ( <. A ,  B >. R
<. C ,  D >.  <->  (
( <. A ,  B >.  e.  ( S  X.  S )  /\  <. C ,  D >.  e.  ( S  X.  S ) )  /\  E. z E. w E. v E. u ( ( <. A ,  B >.  = 
<. z ,  w >.  /\ 
<. C ,  D >.  = 
<. v ,  u >. )  /\  ph ) ) )
19 opbrop.1 . . . . 5  |-  ( ( ( z  =  A  /\  w  =  B )  /\  ( v  =  C  /\  u  =  D ) )  -> 
( ph  <->  ps ) )
2019copsex4g 4691 . . . 4  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( E. z E. w E. v E. u ( ( <. A ,  B >.  = 
<. z ,  w >.  /\ 
<. C ,  D >.  = 
<. v ,  u >. )  /\  ph )  <->  ps )
)
2120anbi2d 703 . . 3  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( ( ( <. A ,  B >.  e.  ( S  X.  S
)  /\  <. C ,  D >.  e.  ( S  X.  S ) )  /\  E. z E. w E. v E. u ( ( <. A ,  B >.  = 
<. z ,  w >.  /\ 
<. C ,  D >.  = 
<. v ,  u >. )  /\  ph ) )  <-> 
( ( <. A ,  B >.  e.  ( S  X.  S )  /\  <. C ,  D >.  e.  ( S  X.  S
) )  /\  ps ) ) )
2218, 21syl5bb 257 . 2  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( <. A ,  B >. R <. C ,  D >.  <-> 
( ( <. A ,  B >.  e.  ( S  X.  S )  /\  <. C ,  D >.  e.  ( S  X.  S
) )  /\  ps ) ) )
23 opelxpi 4982 . . . 4  |-  ( ( A  e.  S  /\  B  e.  S )  -> 
<. A ,  B >.  e.  ( S  X.  S
) )
24 opelxpi 4982 . . . 4  |-  ( ( C  e.  S  /\  D  e.  S )  -> 
<. C ,  D >.  e.  ( S  X.  S
) )
2523, 24anim12i 566 . . 3  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( <. A ,  B >.  e.  ( S  X.  S )  /\  <. C ,  D >.  e.  ( S  X.  S ) ) )
2625biantrurd 508 . 2  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( ps  <->  ( ( <. A ,  B >.  e.  ( S  X.  S
)  /\  <. C ,  D >.  e.  ( S  X.  S ) )  /\  ps ) ) )
2722, 26bitr4d 256 1  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( <. A ,  B >. R <. C ,  D >.  <->  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370   E.wex 1587    e. wcel 1758   <.cop 3994   class class class wbr 4403   {copab 4460    X. cxp 4949
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 4524  ax-nul 4532  ax-pr 4642
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 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-br 4404  df-opab 4462  df-xp 4957
This theorem is referenced by:  ecopoveq  7314  ovec  7323
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