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Theorem brtxp2 24492
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 24489 . . . . . . 7  |-  ( R 
(x)  S )  C_  ( _V  X.  ( _V  X.  _V ) )
21brel 4753 . . . . . 6  |-  ( A ( R  (x)  S
) B  ->  ( A  e.  _V  /\  B  e.  ( _V  X.  _V ) ) )
32simprd 449 . . . . 5  |-  ( A ( R  (x)  S
) B  ->  B  e.  ( _V  X.  _V ) )
4 elvv 4764 . . . . 5  |-  ( B  e.  ( _V  X.  _V )  <->  E. x E. y  B  =  <. x ,  y >. )
53, 4sylib 188 . . . 4  |-  ( A ( R  (x)  S
) B  ->  E. x E. y  B  =  <. x ,  y >.
)
65pm4.71ri 614 . . 3  |-  ( A ( R  (x)  S
) B  <->  ( E. x E. y  B  = 
<. x ,  y >.  /\  A ( R  (x)  S ) B ) )
7 19.41vv 1855 . . 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 243 . 2  |-  ( A ( R  (x)  S
) B  <->  E. x E. y ( B  = 
<. x ,  y >.  /\  A ( R  (x)  S ) B ) )
9 breq2 4043 . . . 4  |-  ( B  =  <. x ,  y
>.  ->  ( A ( R  (x)  S ) B 
<->  A ( R  (x)  S ) <. x ,  y
>. ) )
109pm5.32i 618 . . 3  |-  ( ( B  =  <. x ,  y >.  /\  A
( R  (x)  S
) B )  <->  ( B  =  <. x ,  y
>.  /\  A ( R 
(x)  S ) <.
x ,  y >.
) )
11102exbii 1573 . 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 2804 . . . . . 6  |-  x  e. 
_V
14 vex 2804 . . . . . 6  |-  y  e. 
_V
1512, 13, 14brtxp 24491 . . . . 5  |-  ( A ( R  (x)  S
) <. x ,  y
>. 
<->  ( A R x  /\  A S y ) )
1615anbi2i 675 . . . 4  |-  ( ( B  =  <. x ,  y >.  /\  A
( R  (x)  S
) <. x ,  y
>. )  <->  ( B  = 
<. x ,  y >.  /\  ( A R x  /\  A S y ) ) )
17 3anass 938 . . . 4  |-  ( ( B  =  <. x ,  y >.  /\  A R x  /\  A S y )  <->  ( B  =  <. x ,  y
>.  /\  ( A R x  /\  A S y ) ) )
1816, 17bitr4i 243 . . 3  |-  ( ( B  =  <. x ,  y >.  /\  A
( R  (x)  S
) <. x ,  y
>. )  <->  ( B  = 
<. x ,  y >.  /\  A R x  /\  A S y ) )
19182exbii 1573 . 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 262 1  |-  ( A ( R  (x)  S
) B  <->  E. x E. y ( B  = 
<. x ,  y >.  /\  A R x  /\  A S y ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1531    = wceq 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656   class class class wbr 4039    X. cxp 4703    (x) ctxp 24444
This theorem is referenced by:  elfuns  24525  brsuccf  24551  brrestrict  24559
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-1st 6138  df-2nd 6139  df-txp 24466
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