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Theorem brtxp 24420
Description: Characterize a trinary relationship over a tail cross product. Together with txpss3v 24418, this completely defines membership in a tail cross. (Contributed by Scott Fenton, 31-Mar-2012.)
Hypotheses
Ref Expression
brtxp.1  |-  X  e. 
_V
brtxp.2  |-  Y  e. 
_V
brtxp.3  |-  Z  e. 
_V
Assertion
Ref Expression
brtxp  |-  ( X ( A  (x)  B
) <. Y ,  Z >.  <-> 
( X A Y  /\  X B Z ) )

Proof of Theorem brtxp
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-txp 24395 . . 3  |-  ( A 
(x)  B )  =  ( ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A )  i^i  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  B ) )
21breqi 4029 . 2  |-  ( X ( A  (x)  B
) <. Y ,  Z >.  <-> 
X ( ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A )  i^i  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  B
) ) <. Y ,  Z >. )
3 brin 4070 . 2  |-  ( X ( ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A )  i^i  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  B ) )
<. Y ,  Z >.  <->  ( X ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A ) <. Y ,  Z >.  /\  X ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  B ) <. Y ,  Z >. ) )
4 brtxp.1 . . . . 5  |-  X  e. 
_V
5 opex 4237 . . . . 5  |-  <. Y ,  Z >.  e.  _V
64, 5brco 4852 . . . 4  |-  ( X ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A
) <. Y ,  Z >.  <->  E. y ( X A y  /\  y `' ( 1st  |`  ( _V  X.  _V ) )
<. Y ,  Z >. ) )
7 ancom 437 . . . . . 6  |-  ( ( X A y  /\  y `' ( 1st  |`  ( _V  X.  _V ) )
<. Y ,  Z >. )  <-> 
( y `' ( 1st  |`  ( _V  X.  _V ) ) <. Y ,  Z >.  /\  X A y ) )
8 vex 2791 . . . . . . . . 9  |-  y  e. 
_V
98, 5brcnv 4864 . . . . . . . 8  |-  ( y `' ( 1st  |`  ( _V  X.  _V ) )
<. Y ,  Z >.  <->  <. Y ,  Z >. ( 1st  |`  ( _V  X.  _V ) ) y )
10 brtxp.2 . . . . . . . . . 10  |-  Y  e. 
_V
11 brtxp.3 . . . . . . . . . 10  |-  Z  e. 
_V
1210, 11opelvv 4735 . . . . . . . . 9  |-  <. Y ,  Z >.  e.  ( _V 
X.  _V )
138brres 4961 . . . . . . . . 9  |-  ( <. Y ,  Z >. ( 1st  |`  ( _V  X.  _V ) ) y  <-> 
( <. Y ,  Z >. 1st y  /\  <. Y ,  Z >.  e.  ( _V  X.  _V )
) )
1412, 13mpbiran2 885 . . . . . . . 8  |-  ( <. Y ,  Z >. ( 1st  |`  ( _V  X.  _V ) ) y  <->  <. Y ,  Z >. 1st y )
1510, 11, 8br1steq 24130 . . . . . . . 8  |-  ( <. Y ,  Z >. 1st y  <->  y  =  Y )
169, 14, 153bitri 262 . . . . . . 7  |-  ( y `' ( 1st  |`  ( _V  X.  _V ) )
<. Y ,  Z >.  <->  y  =  Y )
1716anbi1i 676 . . . . . 6  |-  ( ( y `' ( 1st  |`  ( _V  X.  _V ) ) <. Y ,  Z >.  /\  X A
y )  <->  ( y  =  Y  /\  X A y ) )
187, 17bitri 240 . . . . 5  |-  ( ( X A y  /\  y `' ( 1st  |`  ( _V  X.  _V ) )
<. Y ,  Z >. )  <-> 
( y  =  Y  /\  X A y ) )
1918exbii 1569 . . . 4  |-  ( E. y ( X A y  /\  y `' ( 1st  |`  ( _V  X.  _V ) )
<. Y ,  Z >. )  <->  E. y ( y  =  Y  /\  X A y ) )
20 breq2 4027 . . . . 5  |-  ( y  =  Y  ->  ( X A y  <->  X A Y ) )
2110, 20ceqsexv 2823 . . . 4  |-  ( E. y ( y  =  Y  /\  X A y )  <->  X A Y )
226, 19, 213bitri 262 . . 3  |-  ( X ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A
) <. Y ,  Z >.  <-> 
X A Y )
234, 5brco 4852 . . . 4  |-  ( X ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  B
) <. Y ,  Z >.  <->  E. z ( X B z  /\  z `' ( 2nd  |`  ( _V  X.  _V ) )
<. Y ,  Z >. ) )
24 ancom 437 . . . . . 6  |-  ( ( X B z  /\  z `' ( 2nd  |`  ( _V  X.  _V ) )
<. Y ,  Z >. )  <-> 
( z `' ( 2nd  |`  ( _V  X.  _V ) ) <. Y ,  Z >.  /\  X B z ) )
25 vex 2791 . . . . . . . . 9  |-  z  e. 
_V
2625, 5brcnv 4864 . . . . . . . 8  |-  ( z `' ( 2nd  |`  ( _V  X.  _V ) )
<. Y ,  Z >.  <->  <. Y ,  Z >. ( 2nd  |`  ( _V  X.  _V ) ) z )
2725brres 4961 . . . . . . . . 9  |-  ( <. Y ,  Z >. ( 2nd  |`  ( _V  X.  _V ) ) z  <-> 
( <. Y ,  Z >. 2nd z  /\  <. Y ,  Z >.  e.  ( _V  X.  _V )
) )
2812, 27mpbiran2 885 . . . . . . . 8  |-  ( <. Y ,  Z >. ( 2nd  |`  ( _V  X.  _V ) ) z  <->  <. Y ,  Z >. 2nd z )
2910, 11, 25br2ndeq 24131 . . . . . . . 8  |-  ( <. Y ,  Z >. 2nd z  <->  z  =  Z )
3026, 28, 293bitri 262 . . . . . . 7  |-  ( z `' ( 2nd  |`  ( _V  X.  _V ) )
<. Y ,  Z >.  <->  z  =  Z )
3130anbi1i 676 . . . . . 6  |-  ( ( z `' ( 2nd  |`  ( _V  X.  _V ) ) <. Y ,  Z >.  /\  X B
z )  <->  ( z  =  Z  /\  X B z ) )
3224, 31bitri 240 . . . . 5  |-  ( ( X B z  /\  z `' ( 2nd  |`  ( _V  X.  _V ) )
<. Y ,  Z >. )  <-> 
( z  =  Z  /\  X B z ) )
3332exbii 1569 . . . 4  |-  ( E. z ( X B z  /\  z `' ( 2nd  |`  ( _V  X.  _V ) )
<. Y ,  Z >. )  <->  E. z ( z  =  Z  /\  X B z ) )
34 breq2 4027 . . . . 5  |-  ( z  =  Z  ->  ( X B z  <->  X B Z ) )
3511, 34ceqsexv 2823 . . . 4  |-  ( E. z ( z  =  Z  /\  X B z )  <->  X B Z )
3623, 33, 353bitri 262 . . 3  |-  ( X ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  B
) <. Y ,  Z >.  <-> 
X B Z )
3722, 36anbi12i 678 . 2  |-  ( ( X ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A ) <. Y ,  Z >.  /\  X ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  B ) <. Y ,  Z >. )  <-> 
( X A Y  /\  X B Z ) )
382, 3, 373bitri 262 1  |-  ( X ( A  (x)  B
) <. Y ,  Z >.  <-> 
( X A Y  /\  X B Z ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151   <.cop 3643   class class class wbr 4023    X. cxp 4687   `'ccnv 4688    |` cres 4691    o. ccom 4693   1stc1st 6120   2ndc2nd 6121    (x) ctxp 24373
This theorem is referenced by:  brtxp2  24421  pprodss4v  24424  brpprod  24425  brsset  24429  brtxpsd  24434
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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