Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  brtxp Unicode version

Theorem brtxp 24805
Description: Characterize a trinary relationship over a tail cross product. Together with txpss3v 24803, 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 24780 . . 3  |-  ( A 
(x)  B )  =  ( ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A )  i^i  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  B ) )
21breqi 4066 . 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 4107 . 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 4274 . . . . 5  |-  <. Y ,  Z >.  e.  _V
64, 5brco 4889 . . . 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 2825 . . . . . . . . 9  |-  y  e. 
_V
98, 5brcnv 4901 . . . . . . . 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 4772 . . . . . . . . 9  |-  <. Y ,  Z >.  e.  ( _V 
X.  _V )
138brres 4998 . . . . . . . . 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 24515 . . . . . . . 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 1573 . . . 4  |-  ( E. y ( X A y  /\  y `' ( 1st  |`  ( _V  X.  _V ) )
<. Y ,  Z >. )  <->  E. y ( y  =  Y  /\  X A y ) )
20 breq2 4064 . . . . 5  |-  ( y  =  Y  ->  ( X A y  <->  X A Y ) )
2110, 20ceqsexv 2857 . . . 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 4889 . . . 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 2825 . . . . . . . . 9  |-  z  e. 
_V
2625, 5brcnv 4901 . . . . . . . 8  |-  ( z `' ( 2nd  |`  ( _V  X.  _V ) )
<. Y ,  Z >.  <->  <. Y ,  Z >. ( 2nd  |`  ( _V  X.  _V ) ) z )
2725brres 4998 . . . . . . . . 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 24516 . . . . . . . 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 1573 . . . 4  |-  ( E. z ( X B z  /\  z `' ( 2nd  |`  ( _V  X.  _V ) )
<. Y ,  Z >. )  <->  E. z ( z  =  Z  /\  X B z ) )
34 breq2 4064 . . . . 5  |-  ( z  =  Z  ->  ( X B z  <->  X B Z ) )
3511, 34ceqsexv 2857 . . . 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 1532    = wceq 1633    e. wcel 1701   _Vcvv 2822    i^i cin 3185   <.cop 3677   class class class wbr 4060    X. cxp 4724   `'ccnv 4725    |` cres 4728    o. ccom 4730   1stc1st 6162   2ndc2nd 6163    (x) ctxp 24758
This theorem is referenced by:  brtxp2  24806  pprodss4v  24809  brpprod  24810  brsset  24814  brtxpsd  24819
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-fo 5298  df-fv 5300  df-1st 6164  df-2nd 6165  df-txp 24780
  Copyright terms: Public domain W3C validator