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Theorem txpss3v 25723
Description: A tail cross 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 25698 . 2  |-  ( A 
(x)  B )  =  ( ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A )  i^i  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  B ) )
2 inss1 3561 . . 3  |-  ( ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A )  i^i  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  B
) )  C_  ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A )
3 relco 5368 . . . 4  |-  Rel  ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A )
4 vex 2959 . . . . . . . . 9  |-  z  e. 
_V
5 vex 2959 . . . . . . . . 9  |-  y  e. 
_V
64, 5brcnv 5055 . . . . . . . 8  |-  ( z `' ( 1st  |`  ( _V  X.  _V ) ) y  <->  y ( 1st  |`  ( _V  X.  _V ) ) z )
74brres 5152 . . . . . . . . 9  |-  ( y ( 1st  |`  ( _V  X.  _V ) ) z  <->  ( y 1st z  /\  y  e.  ( _V  X.  _V ) ) )
87simprbi 451 . . . . . . . 8  |-  ( y ( 1st  |`  ( _V  X.  _V ) ) z  ->  y  e.  ( _V  X.  _V )
)
96, 8sylbi 188 . . . . . . 7  |-  ( z `' ( 1st  |`  ( _V  X.  _V ) ) y  ->  y  e.  ( _V  X.  _V )
)
109adantl 453 . . . . . 6  |-  ( ( x A z  /\  z `' ( 1st  |`  ( _V  X.  _V ) ) y )  ->  y  e.  ( _V  X.  _V ) )
1110exlimiv 1644 . . . . 5  |-  ( E. z ( x A z  /\  z `' ( 1st  |`  ( _V  X.  _V ) ) y )  ->  y  e.  ( _V  X.  _V ) )
12 vex 2959 . . . . . 6  |-  x  e. 
_V
1312, 5opelco 5044 . . . . 5  |-  ( <.
x ,  y >.  e.  ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A
)  <->  E. z ( x A z  /\  z `' ( 1st  |`  ( _V  X.  _V ) ) y ) )
14 opelxp 4908 . . . . . 6  |-  ( <.
x ,  y >.  e.  ( _V  X.  ( _V  X.  _V ) )  <-> 
( x  e.  _V  /\  y  e.  ( _V 
X.  _V ) ) )
1512, 14mpbiran 885 . . . . 5  |-  ( <.
x ,  y >.  e.  ( _V  X.  ( _V  X.  _V ) )  <-> 
y  e.  ( _V 
X.  _V ) )
1611, 13, 153imtr4i 258 . . . 4  |-  ( <.
x ,  y >.  e.  ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A
)  ->  <. x ,  y >.  e.  ( _V  X.  ( _V  X.  _V ) ) )
173, 16relssi 4967 . . 3  |-  ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A )  C_  ( _V  X.  ( _V  X.  _V ) )
182, 17sstri 3357 . 2  |-  ( ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A )  i^i  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  B
) )  C_  ( _V  X.  ( _V  X.  _V ) )
191, 18eqsstri 3378 1  |-  ( A 
(x)  B )  C_  ( _V  X.  ( _V  X.  _V ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 359   E.wex 1550    e. wcel 1725   _Vcvv 2956    i^i cin 3319    C_ wss 3320   <.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:  txprel  25724  brtxp2  25726  pprodss4v  25729
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
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-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-br 4213  df-opab 4267  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-res 4890  df-txp 25698
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