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Theorem brtxp2 28051
Description: The binary relationship over a tail cross when the second argument is not an ordered pair. (Contributed by Scott Fenton, 14-Apr-2014.) (Revised by Mario Carneiro, 3-May-2015.)
Hypothesis
Ref Expression
brtxp2.1  |-  A  e. 
_V
Assertion
Ref Expression
brtxp2  |-  ( A ( R  (x)  S
) B  <->  E. x E. y ( B  = 
<. x ,  y >.  /\  A R x  /\  A S y ) )
Distinct variable groups:    x, A, y    x, B, y    x, R, y    x, S, y

Proof of Theorem brtxp2
StepHypRef Expression
1 txpss3v 28048 . . . . . . 7  |-  ( R 
(x)  S )  C_  ( _V  X.  ( _V  X.  _V ) )
21brel 4990 . . . . . 6  |-  ( A ( R  (x)  S
) B  ->  ( A  e.  _V  /\  B  e.  ( _V  X.  _V ) ) )
32simprd 463 . . . . 5  |-  ( A ( R  (x)  S
) B  ->  B  e.  ( _V  X.  _V ) )
4 elvv 5000 . . . . 5  |-  ( B  e.  ( _V  X.  _V )  <->  E. x E. y  B  =  <. x ,  y >. )
53, 4sylib 196 . . . 4  |-  ( A ( R  (x)  S
) B  ->  E. x E. y  B  =  <. x ,  y >.
)
65pm4.71ri 633 . . 3  |-  ( A ( R  (x)  S
) B  <->  ( E. x E. y  B  = 
<. x ,  y >.  /\  A ( R  (x)  S ) B ) )
7 19.41vv 1932 . . 3  |-  ( E. x E. y ( B  =  <. x ,  y >.  /\  A
( R  (x)  S
) B )  <->  ( E. x E. y  B  = 
<. x ,  y >.  /\  A ( R  (x)  S ) B ) )
86, 7bitr4i 252 . 2  |-  ( A ( R  (x)  S
) B  <->  E. x E. y ( B  = 
<. x ,  y >.  /\  A ( R  (x)  S ) B ) )
9 breq2 4399 . . . 4  |-  ( B  =  <. x ,  y
>.  ->  ( A ( R  (x)  S ) B 
<->  A ( R  (x)  S ) <. x ,  y
>. ) )
109pm5.32i 637 . . 3  |-  ( ( B  =  <. x ,  y >.  /\  A
( R  (x)  S
) B )  <->  ( B  =  <. x ,  y
>.  /\  A ( R 
(x)  S ) <.
x ,  y >.
) )
11102exbii 1636 . 2  |-  ( E. x E. y ( B  =  <. x ,  y >.  /\  A
( R  (x)  S
) B )  <->  E. x E. y ( B  = 
<. x ,  y >.  /\  A ( R  (x)  S ) <. x ,  y
>. ) )
12 brtxp2.1 . . . . . 6  |-  A  e. 
_V
13 vex 3075 . . . . . 6  |-  x  e. 
_V
14 vex 3075 . . . . . 6  |-  y  e. 
_V
1512, 13, 14brtxp 28050 . . . . 5  |-  ( A ( R  (x)  S
) <. x ,  y
>. 
<->  ( A R x  /\  A S y ) )
1615anbi2i 694 . . . 4  |-  ( ( B  =  <. x ,  y >.  /\  A
( R  (x)  S
) <. x ,  y
>. )  <->  ( B  = 
<. x ,  y >.  /\  ( A R x  /\  A S y ) ) )
17 3anass 969 . . . 4  |-  ( ( B  =  <. x ,  y >.  /\  A R x  /\  A S y )  <->  ( B  =  <. x ,  y
>.  /\  ( A R x  /\  A S y ) ) )
1816, 17bitr4i 252 . . 3  |-  ( ( B  =  <. x ,  y >.  /\  A
( R  (x)  S
) <. x ,  y
>. )  <->  ( B  = 
<. x ,  y >.  /\  A R x  /\  A S y ) )
19182exbii 1636 . 2  |-  ( E. x E. y ( B  =  <. x ,  y >.  /\  A
( R  (x)  S
) <. x ,  y
>. )  <->  E. x E. y
( B  =  <. x ,  y >.  /\  A R x  /\  A S y ) )
208, 11, 193bitri 271 1  |-  ( A ( R  (x)  S
) B  <->  E. x E. y ( B  = 
<. x ,  y >.  /\  A R x  /\  A S y ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370   E.wex 1587    e. wcel 1758   _Vcvv 3072   <.cop 3986   class class class wbr 4395    X. cxp 4941    (x) ctxp 27999
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-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
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 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-fo 5527  df-fv 5529  df-1st 6682  df-2nd 6683  df-txp 28023
This theorem is referenced by:  brsuccf  28111  brrestrict  28119
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