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Theorem brtxp2 24421
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 24418 . . . . . . 7  |-  ( R 
(x)  S )  C_  ( _V  X.  ( _V  X.  _V ) )
21brel 4737 . . . . . 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 4748 . . . . 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 1843 . . 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 4027 . . . 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 1570 . 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 2791 . . . . . 6  |-  x  e. 
_V
14 vex 2791 . . . . . 6  |-  y  e. 
_V
1512, 13, 14brtxp 24420 . . . . 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 1570 . 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 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643   class class class wbr 4023    X. cxp 4687    (x) ctxp 24373
This theorem is referenced by:  elfuns  24454  brsuccf  24480  brrestrict  24487
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-1st 6122  df-2nd 6123  df-txp 24395
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