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Theorem brtxp2 29684
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 29681 . . . . . . 7  |-  ( R 
(x)  S )  C_  ( _V  X.  ( _V  X.  _V ) )
21brel 4962 . . . . . 6  |-  ( A ( R  (x)  S
) B  ->  ( A  e.  _V  /\  B  e.  ( _V  X.  _V ) ) )
32simprd 461 . . . . 5  |-  ( A ( R  (x)  S
) B  ->  B  e.  ( _V  X.  _V ) )
4 elvv 4972 . . . . 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 631 . . 3  |-  ( A ( R  (x)  S
) B  <->  ( E. x E. y  B  = 
<. x ,  y >.  /\  A ( R  (x)  S ) B ) )
7 19.41vv 1780 . . 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 4371 . . . 4  |-  ( B  =  <. x ,  y
>.  ->  ( A ( R  (x)  S ) B 
<->  A ( R  (x)  S ) <. x ,  y
>. ) )
109pm5.32i 635 . . 3  |-  ( ( B  =  <. x ,  y >.  /\  A
( R  (x)  S
) B )  <->  ( B  =  <. x ,  y
>.  /\  A ( R 
(x)  S ) <.
x ,  y >.
) )
11102exbii 1676 . 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 3037 . . . . . 6  |-  x  e. 
_V
14 vex 3037 . . . . . 6  |-  y  e. 
_V
1512, 13, 14brtxp 29683 . . . . 5  |-  ( A ( R  (x)  S
) <. x ,  y
>. 
<->  ( A R x  /\  A S y ) )
1615anbi2i 692 . . . 4  |-  ( ( B  =  <. x ,  y >.  /\  A
( R  (x)  S
) <. x ,  y
>. )  <->  ( B  = 
<. x ,  y >.  /\  ( A R x  /\  A S y ) ) )
17 3anass 975 . . . 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 1676 . 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 367    /\ w3a 971    = wceq 1399   E.wex 1620    e. wcel 1826   _Vcvv 3034   <.cop 3950   class class class wbr 4367    X. cxp 4911    (x) ctxp 29632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-fo 5502  df-fv 5504  df-1st 6699  df-2nd 6700  df-txp 29656
This theorem is referenced by:  brsuccf  29744  brrestrict  29752
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