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Theorem brtxp 25725
Description: Characterize a trinary relationship over a tail cross product. Together with txpss3v 25723, 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 25698 . . 3  |-  ( A 
(x)  B )  =  ( ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A )  i^i  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  B ) )
21breqi 4218 . 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 4259 . 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 4427 . . . . 5  |-  <. Y ,  Z >.  e.  _V
64, 5brco 5043 . . . 4  |-  ( X ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A
) <. Y ,  Z >.  <->  E. y ( X A y  /\  y `' ( 1st  |`  ( _V  X.  _V ) )
<. Y ,  Z >. ) )
7 ancom 438 . . . . . 6  |-  ( ( X A y  /\  y `' ( 1st  |`  ( _V  X.  _V ) )
<. Y ,  Z >. )  <-> 
( y `' ( 1st  |`  ( _V  X.  _V ) ) <. Y ,  Z >.  /\  X A y ) )
8 vex 2959 . . . . . . . . 9  |-  y  e. 
_V
98, 5brcnv 5055 . . . . . . . 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 4924 . . . . . . . . 9  |-  <. Y ,  Z >.  e.  ( _V 
X.  _V )
138brres 5152 . . . . . . . . 9  |-  ( <. Y ,  Z >. ( 1st  |`  ( _V  X.  _V ) ) y  <-> 
( <. Y ,  Z >. 1st y  /\  <. Y ,  Z >.  e.  ( _V  X.  _V )
) )
1412, 13mpbiran2 886 . . . . . . . 8  |-  ( <. Y ,  Z >. ( 1st  |`  ( _V  X.  _V ) ) y  <->  <. Y ,  Z >. 1st y )
1510, 11, 8br1steq 25398 . . . . . . . 8  |-  ( <. Y ,  Z >. 1st y  <->  y  =  Y )
169, 14, 153bitri 263 . . . . . . 7  |-  ( y `' ( 1st  |`  ( _V  X.  _V ) )
<. Y ,  Z >.  <->  y  =  Y )
1716anbi1i 677 . . . . . 6  |-  ( ( y `' ( 1st  |`  ( _V  X.  _V ) ) <. Y ,  Z >.  /\  X A
y )  <->  ( y  =  Y  /\  X A y ) )
187, 17bitri 241 . . . . 5  |-  ( ( X A y  /\  y `' ( 1st  |`  ( _V  X.  _V ) )
<. Y ,  Z >. )  <-> 
( y  =  Y  /\  X A y ) )
1918exbii 1592 . . . 4  |-  ( E. y ( X A y  /\  y `' ( 1st  |`  ( _V  X.  _V ) )
<. Y ,  Z >. )  <->  E. y ( y  =  Y  /\  X A y ) )
20 breq2 4216 . . . . 5  |-  ( y  =  Y  ->  ( X A y  <->  X A Y ) )
2110, 20ceqsexv 2991 . . . 4  |-  ( E. y ( y  =  Y  /\  X A y )  <->  X A Y )
226, 19, 213bitri 263 . . 3  |-  ( X ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A
) <. Y ,  Z >.  <-> 
X A Y )
234, 5brco 5043 . . . 4  |-  ( X ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  B
) <. Y ,  Z >.  <->  E. z ( X B z  /\  z `' ( 2nd  |`  ( _V  X.  _V ) )
<. Y ,  Z >. ) )
24 ancom 438 . . . . . 6  |-  ( ( X B z  /\  z `' ( 2nd  |`  ( _V  X.  _V ) )
<. Y ,  Z >. )  <-> 
( z `' ( 2nd  |`  ( _V  X.  _V ) ) <. Y ,  Z >.  /\  X B z ) )
25 vex 2959 . . . . . . . . 9  |-  z  e. 
_V
2625, 5brcnv 5055 . . . . . . . 8  |-  ( z `' ( 2nd  |`  ( _V  X.  _V ) )
<. Y ,  Z >.  <->  <. Y ,  Z >. ( 2nd  |`  ( _V  X.  _V ) ) z )
2725brres 5152 . . . . . . . . 9  |-  ( <. Y ,  Z >. ( 2nd  |`  ( _V  X.  _V ) ) z  <-> 
( <. Y ,  Z >. 2nd z  /\  <. Y ,  Z >.  e.  ( _V  X.  _V )
) )
2812, 27mpbiran2 886 . . . . . . . 8  |-  ( <. Y ,  Z >. ( 2nd  |`  ( _V  X.  _V ) ) z  <->  <. Y ,  Z >. 2nd z )
2910, 11, 25br2ndeq 25399 . . . . . . . 8  |-  ( <. Y ,  Z >. 2nd z  <->  z  =  Z )
3026, 28, 293bitri 263 . . . . . . 7  |-  ( z `' ( 2nd  |`  ( _V  X.  _V ) )
<. Y ,  Z >.  <->  z  =  Z )
3130anbi1i 677 . . . . . 6  |-  ( ( z `' ( 2nd  |`  ( _V  X.  _V ) ) <. Y ,  Z >.  /\  X B
z )  <->  ( z  =  Z  /\  X B z ) )
3224, 31bitri 241 . . . . 5  |-  ( ( X B z  /\  z `' ( 2nd  |`  ( _V  X.  _V ) )
<. Y ,  Z >. )  <-> 
( z  =  Z  /\  X B z ) )
3332exbii 1592 . . . 4  |-  ( E. z ( X B z  /\  z `' ( 2nd  |`  ( _V  X.  _V ) )
<. Y ,  Z >. )  <->  E. z ( z  =  Z  /\  X B z ) )
34 breq2 4216 . . . . 5  |-  ( z  =  Z  ->  ( X B z  <->  X B Z ) )
3511, 34ceqsexv 2991 . . . 4  |-  ( E. z ( z  =  Z  /\  X B z )  <->  X B Z )
3623, 33, 353bitri 263 . . 3  |-  ( X ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  B
) <. Y ,  Z >.  <-> 
X B Z )
3722, 36anbi12i 679 . 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 263 1  |-  ( X ( A  (x)  B
) <. Y ,  Z >.  <-> 
( X A Y  /\  X B Z ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   _Vcvv 2956    i^i cin 3319   <.cop 3817   class class class wbr 4212    X. cxp 4876   `'ccnv 4877    |` cres 4880    o. ccom 4882   1stc1st 6347   2ndc2nd 6348    (x) ctxp 25674
This theorem is referenced by:  brtxp2  25726  pprodss4v  25729  brpprod  25730  brsset  25734  brtxpsd  25739  elfuns  25760
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fo 5460  df-fv 5462  df-1st 6349  df-2nd 6350  df-txp 25698
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