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Theorem brtxp 29383
Description: Characterize a trinary relationship over a tail Cartesian product. Together with txpss3v 29381, 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 29356 . . 3  |-  ( A 
(x)  B )  =  ( ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A )  i^i  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  B ) )
21breqi 4453 . 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 4496 . 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 4711 . . . . 5  |-  <. Y ,  Z >.  e.  _V
64, 5brco 5173 . . . 4  |-  ( X ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A
) <. Y ,  Z >.  <->  E. y ( X A y  /\  y `' ( 1st  |`  ( _V  X.  _V ) )
<. Y ,  Z >. ) )
7 ancom 450 . . . . . 6  |-  ( ( X A y  /\  y `' ( 1st  |`  ( _V  X.  _V ) )
<. Y ,  Z >. )  <-> 
( y `' ( 1st  |`  ( _V  X.  _V ) ) <. Y ,  Z >.  /\  X A y ) )
8 vex 3116 . . . . . . . . 9  |-  y  e. 
_V
98, 5brcnv 5185 . . . . . . . 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 5046 . . . . . . . . 9  |-  <. Y ,  Z >.  e.  ( _V 
X.  _V )
138brres 5280 . . . . . . . . 9  |-  ( <. Y ,  Z >. ( 1st  |`  ( _V  X.  _V ) ) y  <-> 
( <. Y ,  Z >. 1st y  /\  <. Y ,  Z >.  e.  ( _V  X.  _V )
) )
1412, 13mpbiran2 917 . . . . . . . 8  |-  ( <. Y ,  Z >. ( 1st  |`  ( _V  X.  _V ) ) y  <->  <. Y ,  Z >. 1st y )
1510, 11, 8br1steq 29057 . . . . . . . 8  |-  ( <. Y ,  Z >. 1st y  <->  y  =  Y )
169, 14, 153bitri 271 . . . . . . 7  |-  ( y `' ( 1st  |`  ( _V  X.  _V ) )
<. Y ,  Z >.  <->  y  =  Y )
1716anbi1i 695 . . . . . 6  |-  ( ( y `' ( 1st  |`  ( _V  X.  _V ) ) <. Y ,  Z >.  /\  X A
y )  <->  ( y  =  Y  /\  X A y ) )
187, 17bitri 249 . . . . 5  |-  ( ( X A y  /\  y `' ( 1st  |`  ( _V  X.  _V ) )
<. Y ,  Z >. )  <-> 
( y  =  Y  /\  X A y ) )
1918exbii 1644 . . . 4  |-  ( E. y ( X A y  /\  y `' ( 1st  |`  ( _V  X.  _V ) )
<. Y ,  Z >. )  <->  E. y ( y  =  Y  /\  X A y ) )
20 breq2 4451 . . . . 5  |-  ( y  =  Y  ->  ( X A y  <->  X A Y ) )
2110, 20ceqsexv 3150 . . . 4  |-  ( E. y ( y  =  Y  /\  X A y )  <->  X A Y )
226, 19, 213bitri 271 . . 3  |-  ( X ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A
) <. Y ,  Z >.  <-> 
X A Y )
234, 5brco 5173 . . . 4  |-  ( X ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  B
) <. Y ,  Z >.  <->  E. z ( X B z  /\  z `' ( 2nd  |`  ( _V  X.  _V ) )
<. Y ,  Z >. ) )
24 ancom 450 . . . . . 6  |-  ( ( X B z  /\  z `' ( 2nd  |`  ( _V  X.  _V ) )
<. Y ,  Z >. )  <-> 
( z `' ( 2nd  |`  ( _V  X.  _V ) ) <. Y ,  Z >.  /\  X B z ) )
25 vex 3116 . . . . . . . . 9  |-  z  e. 
_V
2625, 5brcnv 5185 . . . . . . . 8  |-  ( z `' ( 2nd  |`  ( _V  X.  _V ) )
<. Y ,  Z >.  <->  <. Y ,  Z >. ( 2nd  |`  ( _V  X.  _V ) ) z )
2725brres 5280 . . . . . . . . 9  |-  ( <. Y ,  Z >. ( 2nd  |`  ( _V  X.  _V ) ) z  <-> 
( <. Y ,  Z >. 2nd z  /\  <. Y ,  Z >.  e.  ( _V  X.  _V )
) )
2812, 27mpbiran2 917 . . . . . . . 8  |-  ( <. Y ,  Z >. ( 2nd  |`  ( _V  X.  _V ) ) z  <->  <. Y ,  Z >. 2nd z )
2910, 11, 25br2ndeq 29058 . . . . . . . 8  |-  ( <. Y ,  Z >. 2nd z  <->  z  =  Z )
3026, 28, 293bitri 271 . . . . . . 7  |-  ( z `' ( 2nd  |`  ( _V  X.  _V ) )
<. Y ,  Z >.  <->  z  =  Z )
3130anbi1i 695 . . . . . 6  |-  ( ( z `' ( 2nd  |`  ( _V  X.  _V ) ) <. Y ,  Z >.  /\  X B
z )  <->  ( z  =  Z  /\  X B z ) )
3224, 31bitri 249 . . . . 5  |-  ( ( X B z  /\  z `' ( 2nd  |`  ( _V  X.  _V ) )
<. Y ,  Z >. )  <-> 
( z  =  Z  /\  X B z ) )
3332exbii 1644 . . . 4  |-  ( E. z ( X B z  /\  z `' ( 2nd  |`  ( _V  X.  _V ) )
<. Y ,  Z >. )  <->  E. z ( z  =  Z  /\  X B z ) )
34 breq2 4451 . . . . 5  |-  ( z  =  Z  ->  ( X B z  <->  X B Z ) )
3511, 34ceqsexv 3150 . . . 4  |-  ( E. z ( z  =  Z  /\  X B z )  <->  X B Z )
3623, 33, 353bitri 271 . . 3  |-  ( X ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  B
) <. Y ,  Z >.  <-> 
X B Z )
3722, 36anbi12i 697 . 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 271 1  |-  ( X ( A  (x)  B
) <. Y ,  Z >.  <-> 
( X A Y  /\  X B Z ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767   _Vcvv 3113    i^i cin 3475   <.cop 4033   class class class wbr 4447    X. cxp 4997   `'ccnv 4998    |` cres 5001    o. ccom 5003   1stc1st 6783   2ndc2nd 6784    (x) ctxp 29332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fo 5594  df-fv 5596  df-1st 6785  df-2nd 6786  df-txp 29356
This theorem is referenced by:  brtxp2  29384  pprodss4v  29387  brpprod  29388  brsset  29392  brtxpsd  29397  elfuns  29418
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