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Theorem brtxp 30640
Description: Characterize a trinary relationship over a tail Cartesian product. Together with txpss3v 30638, 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 30613 . . 3  |-  ( A 
(x)  B )  =  ( ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A )  i^i  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  B ) )
21breqi 4407 . 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 4451 . 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 4663 . . . . 5  |-  <. Y ,  Z >.  e.  _V
64, 5brco 5004 . . . 4  |-  ( X ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A
) <. Y ,  Z >.  <->  E. y ( X A y  /\  y `' ( 1st  |`  ( _V  X.  _V ) )
<. Y ,  Z >. ) )
7 ancom 452 . . . . . 6  |-  ( ( X A y  /\  y `' ( 1st  |`  ( _V  X.  _V ) )
<. Y ,  Z >. )  <-> 
( y `' ( 1st  |`  ( _V  X.  _V ) ) <. Y ,  Z >.  /\  X A y ) )
8 vex 3047 . . . . . . . . 9  |-  y  e. 
_V
98, 5brcnv 5016 . . . . . . . 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 4880 . . . . . . . . 9  |-  <. Y ,  Z >.  e.  ( _V 
X.  _V )
138brres 5110 . . . . . . . . 9  |-  ( <. Y ,  Z >. ( 1st  |`  ( _V  X.  _V ) ) y  <-> 
( <. Y ,  Z >. 1st y  /\  <. Y ,  Z >.  e.  ( _V  X.  _V )
) )
1412, 13mpbiran2 929 . . . . . . . 8  |-  ( <. Y ,  Z >. ( 1st  |`  ( _V  X.  _V ) ) y  <->  <. Y ,  Z >. 1st y )
1510, 11, 8br1steq 30407 . . . . . . . 8  |-  ( <. Y ,  Z >. 1st y  <->  y  =  Y )
169, 14, 153bitri 275 . . . . . . 7  |-  ( y `' ( 1st  |`  ( _V  X.  _V ) )
<. Y ,  Z >.  <->  y  =  Y )
1716anbi1i 700 . . . . . 6  |-  ( ( y `' ( 1st  |`  ( _V  X.  _V ) ) <. Y ,  Z >.  /\  X A
y )  <->  ( y  =  Y  /\  X A y ) )
187, 17bitri 253 . . . . 5  |-  ( ( X A y  /\  y `' ( 1st  |`  ( _V  X.  _V ) )
<. Y ,  Z >. )  <-> 
( y  =  Y  /\  X A y ) )
1918exbii 1717 . . . 4  |-  ( E. y ( X A y  /\  y `' ( 1st  |`  ( _V  X.  _V ) )
<. Y ,  Z >. )  <->  E. y ( y  =  Y  /\  X A y ) )
20 breq2 4405 . . . . 5  |-  ( y  =  Y  ->  ( X A y  <->  X A Y ) )
2110, 20ceqsexv 3083 . . . 4  |-  ( E. y ( y  =  Y  /\  X A y )  <->  X A Y )
226, 19, 213bitri 275 . . 3  |-  ( X ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A
) <. Y ,  Z >.  <-> 
X A Y )
234, 5brco 5004 . . . 4  |-  ( X ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  B
) <. Y ,  Z >.  <->  E. z ( X B z  /\  z `' ( 2nd  |`  ( _V  X.  _V ) )
<. Y ,  Z >. ) )
24 ancom 452 . . . . . 6  |-  ( ( X B z  /\  z `' ( 2nd  |`  ( _V  X.  _V ) )
<. Y ,  Z >. )  <-> 
( z `' ( 2nd  |`  ( _V  X.  _V ) ) <. Y ,  Z >.  /\  X B z ) )
25 vex 3047 . . . . . . . . 9  |-  z  e. 
_V
2625, 5brcnv 5016 . . . . . . . 8  |-  ( z `' ( 2nd  |`  ( _V  X.  _V ) )
<. Y ,  Z >.  <->  <. Y ,  Z >. ( 2nd  |`  ( _V  X.  _V ) ) z )
2725brres 5110 . . . . . . . . 9  |-  ( <. Y ,  Z >. ( 2nd  |`  ( _V  X.  _V ) ) z  <-> 
( <. Y ,  Z >. 2nd z  /\  <. Y ,  Z >.  e.  ( _V  X.  _V )
) )
2812, 27mpbiran2 929 . . . . . . . 8  |-  ( <. Y ,  Z >. ( 2nd  |`  ( _V  X.  _V ) ) z  <->  <. Y ,  Z >. 2nd z )
2910, 11, 25br2ndeq 30408 . . . . . . . 8  |-  ( <. Y ,  Z >. 2nd z  <->  z  =  Z )
3026, 28, 293bitri 275 . . . . . . 7  |-  ( z `' ( 2nd  |`  ( _V  X.  _V ) )
<. Y ,  Z >.  <->  z  =  Z )
3130anbi1i 700 . . . . . 6  |-  ( ( z `' ( 2nd  |`  ( _V  X.  _V ) ) <. Y ,  Z >.  /\  X B
z )  <->  ( z  =  Z  /\  X B z ) )
3224, 31bitri 253 . . . . 5  |-  ( ( X B z  /\  z `' ( 2nd  |`  ( _V  X.  _V ) )
<. Y ,  Z >. )  <-> 
( z  =  Z  /\  X B z ) )
3332exbii 1717 . . . 4  |-  ( E. z ( X B z  /\  z `' ( 2nd  |`  ( _V  X.  _V ) )
<. Y ,  Z >. )  <->  E. z ( z  =  Z  /\  X B z ) )
34 breq2 4405 . . . . 5  |-  ( z  =  Z  ->  ( X B z  <->  X B Z ) )
3511, 34ceqsexv 3083 . . . 4  |-  ( E. z ( z  =  Z  /\  X B z )  <->  X B Z )
3623, 33, 353bitri 275 . . 3  |-  ( X ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  B
) <. Y ,  Z >.  <-> 
X B Z )
3722, 36anbi12i 702 . 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 275 1  |-  ( X ( A  (x)  B
) <. Y ,  Z >.  <-> 
( X A Y  /\  X B Z ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    /\ wa 371    = wceq 1443   E.wex 1662    e. wcel 1886   _Vcvv 3044    i^i cin 3402   <.cop 3973   class class class wbr 4401    X. cxp 4831   `'ccnv 4832    |` cres 4835    o. ccom 4837   1stc1st 6788   2ndc2nd 6789    (x) ctxp 30589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-sbc 3267  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-fo 5587  df-fv 5589  df-1st 6790  df-2nd 6791  df-txp 30613
This theorem is referenced by:  brtxp2  30641  pprodss4v  30644  brpprod  30645  brsset  30649  brtxpsd  30654  elfuns  30675
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