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Theorem brtxp2 30697
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 30694 . . . . . . 7  |-  ( R 
(x)  S )  C_  ( _V  X.  ( _V  X.  _V ) )
21brel 4902 . . . . . 6  |-  ( A ( R  (x)  S
) B  ->  ( A  e.  _V  /\  B  e.  ( _V  X.  _V ) ) )
32simprd 469 . . . . 5  |-  ( A ( R  (x)  S
) B  ->  B  e.  ( _V  X.  _V ) )
4 elvv 4912 . . . . 5  |-  ( B  e.  ( _V  X.  _V )  <->  E. x E. y  B  =  <. x ,  y >. )
53, 4sylib 201 . . . 4  |-  ( A ( R  (x)  S
) B  ->  E. x E. y  B  =  <. x ,  y >.
)
65pm4.71ri 643 . . 3  |-  ( A ( R  (x)  S
) B  <->  ( E. x E. y  B  = 
<. x ,  y >.  /\  A ( R  (x)  S ) B ) )
7 19.41vv 1842 . . 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 260 . 2  |-  ( A ( R  (x)  S
) B  <->  E. x E. y ( B  = 
<. x ,  y >.  /\  A ( R  (x)  S ) B ) )
9 breq2 4420 . . . 4  |-  ( B  =  <. x ,  y
>.  ->  ( A ( R  (x)  S ) B 
<->  A ( R  (x)  S ) <. x ,  y
>. ) )
109pm5.32i 647 . . 3  |-  ( ( B  =  <. x ,  y >.  /\  A
( R  (x)  S
) B )  <->  ( B  =  <. x ,  y
>.  /\  A ( R 
(x)  S ) <.
x ,  y >.
) )
11102exbii 1730 . 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 3060 . . . . . 6  |-  x  e. 
_V
14 vex 3060 . . . . . 6  |-  y  e. 
_V
1512, 13, 14brtxp 30696 . . . . 5  |-  ( A ( R  (x)  S
) <. x ,  y
>. 
<->  ( A R x  /\  A S y ) )
1615anbi2i 705 . . . 4  |-  ( ( B  =  <. x ,  y >.  /\  A
( R  (x)  S
) <. x ,  y
>. )  <->  ( B  = 
<. x ,  y >.  /\  ( A R x  /\  A S y ) ) )
17 3anass 995 . . . 4  |-  ( ( B  =  <. x ,  y >.  /\  A R x  /\  A S y )  <->  ( B  =  <. x ,  y
>.  /\  ( A R x  /\  A S y ) ) )
1816, 17bitr4i 260 . . 3  |-  ( ( B  =  <. x ,  y >.  /\  A
( R  (x)  S
) <. x ,  y
>. )  <->  ( B  = 
<. x ,  y >.  /\  A R x  /\  A S y ) )
19182exbii 1730 . 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 279 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 189    /\ wa 375    /\ w3a 991    = wceq 1455   E.wex 1674    e. wcel 1898   _Vcvv 3057   <.cop 3986   class class class wbr 4416    X. cxp 4851    (x) ctxp 30645
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fo 5607  df-fv 5609  df-1st 6820  df-2nd 6821  df-txp 30669
This theorem is referenced by:  brsuccf  30757  brrestrict  30765
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