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Theorem txpss3v 27931
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 27906 . 2  |-  ( A 
(x)  B )  =  ( ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A )  i^i  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  B ) )
2 inss1 3591 . . 3  |-  ( ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A )  i^i  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  B
) )  C_  ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A )
3 relco 5357 . . . 4  |-  Rel  ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A )
4 vex 2996 . . . . . . . . 9  |-  z  e. 
_V
5 vex 2996 . . . . . . . . 9  |-  y  e. 
_V
64, 5brcnv 5043 . . . . . . . 8  |-  ( z `' ( 1st  |`  ( _V  X.  _V ) ) y  <->  y ( 1st  |`  ( _V  X.  _V ) ) z )
74brres 5138 . . . . . . . . 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 1688 . . . . 5  |-  ( E. z ( x A z  /\  z `' ( 1st  |`  ( _V  X.  _V ) ) y )  ->  y  e.  ( _V  X.  _V ) )
12 vex 2996 . . . . . 6  |-  x  e. 
_V
1312, 5opelco 5032 . . . . 5  |-  ( <.
x ,  y >.  e.  ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A
)  <->  E. z ( x A z  /\  z `' ( 1st  |`  ( _V  X.  _V ) ) y ) )
14 opelxp 4890 . . . . . 6  |-  ( <.
x ,  y >.  e.  ( _V  X.  ( _V  X.  _V ) )  <-> 
( x  e.  _V  /\  y  e.  ( _V 
X.  _V ) ) )
1512, 14mpbiran 909 . . . . 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 4952 . . 3  |-  ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A )  C_  ( _V  X.  ( _V  X.  _V ) )
182, 17sstri 3386 . 2  |-  ( ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A )  i^i  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  B
) )  C_  ( _V  X.  ( _V  X.  _V ) )
191, 18eqsstri 3407 1  |-  ( A 
(x)  B )  C_  ( _V  X.  ( _V  X.  _V ) )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369   E.wex 1586    e. wcel 1756   _Vcvv 2993    i^i cin 3348    C_ wss 3349   <.cop 3904   class class class wbr 4313    X. cxp 4859   `'ccnv 4860    |` cres 4863    o. ccom 4865   1stc1st 6596   2ndc2nd 6597    (x) ctxp 27882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pr 4552
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-br 4314  df-opab 4372  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-res 4873  df-txp 27906
This theorem is referenced by:  txprel  27932  brtxp2  27934  pprodss4v  27937
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