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Theorem txpss3v 29690
Description: A tail Cartesian product is a subset of the class of ordered triples. (Contributed by Scott Fenton, 31-Mar-2012.)
Assertion
Ref Expression
txpss3v  |-  ( A 
(x)  B )  C_  ( _V  X.  ( _V  X.  _V ) )

Proof of Theorem txpss3v
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-txp 29665 . 2  |-  ( A 
(x)  B )  =  ( ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A )  i^i  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  B ) )
2 inss1 3714 . . 3  |-  ( ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A )  i^i  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  B
) )  C_  ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A )
3 relco 5511 . . . 4  |-  Rel  ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A )
4 vex 3112 . . . . . . . . 9  |-  z  e. 
_V
5 vex 3112 . . . . . . . . 9  |-  y  e. 
_V
64, 5brcnv 5195 . . . . . . . 8  |-  ( z `' ( 1st  |`  ( _V  X.  _V ) ) y  <->  y ( 1st  |`  ( _V  X.  _V ) ) z )
74brres 5290 . . . . . . . . 9  |-  ( y ( 1st  |`  ( _V  X.  _V ) ) z  <->  ( y 1st z  /\  y  e.  ( _V  X.  _V ) ) )
87simprbi 464 . . . . . . . 8  |-  ( y ( 1st  |`  ( _V  X.  _V ) ) z  ->  y  e.  ( _V  X.  _V )
)
96, 8sylbi 195 . . . . . . 7  |-  ( z `' ( 1st  |`  ( _V  X.  _V ) ) y  ->  y  e.  ( _V  X.  _V )
)
109adantl 466 . . . . . 6  |-  ( ( x A z  /\  z `' ( 1st  |`  ( _V  X.  _V ) ) y )  ->  y  e.  ( _V  X.  _V ) )
1110exlimiv 1723 . . . . 5  |-  ( E. z ( x A z  /\  z `' ( 1st  |`  ( _V  X.  _V ) ) y )  ->  y  e.  ( _V  X.  _V ) )
12 vex 3112 . . . . . 6  |-  x  e. 
_V
1312, 5opelco 5184 . . . . 5  |-  ( <.
x ,  y >.  e.  ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A
)  <->  E. z ( x A z  /\  z `' ( 1st  |`  ( _V  X.  _V ) ) y ) )
14 opelxp 5038 . . . . . 6  |-  ( <.
x ,  y >.  e.  ( _V  X.  ( _V  X.  _V ) )  <-> 
( x  e.  _V  /\  y  e.  ( _V 
X.  _V ) ) )
1512, 14mpbiran 918 . . . . 5  |-  ( <.
x ,  y >.  e.  ( _V  X.  ( _V  X.  _V ) )  <-> 
y  e.  ( _V 
X.  _V ) )
1611, 13, 153imtr4i 266 . . . 4  |-  ( <.
x ,  y >.  e.  ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A
)  ->  <. x ,  y >.  e.  ( _V  X.  ( _V  X.  _V ) ) )
173, 16relssi 5103 . . 3  |-  ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A )  C_  ( _V  X.  ( _V  X.  _V ) )
182, 17sstri 3508 . 2  |-  ( ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A )  i^i  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  B
) )  C_  ( _V  X.  ( _V  X.  _V ) )
191, 18eqsstri 3529 1  |-  ( A 
(x)  B )  C_  ( _V  X.  ( _V  X.  _V ) )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369   E.wex 1613    e. wcel 1819   _Vcvv 3109    i^i cin 3470    C_ wss 3471   <.cop 4038   class class class wbr 4456    X. cxp 5006   `'ccnv 5007    |` cres 5010    o. ccom 5012   1stc1st 6797   2ndc2nd 6798    (x) ctxp 29641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-res 5020  df-txp 29665
This theorem is referenced by:  txprel  29691  brtxp2  29693  pprodss4v  29696
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