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Theorem txpss3v 25632
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 25609 . 2  |-  ( A 
(x)  B )  =  ( ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A )  i^i  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  B ) )
2 inss1 3521 . . 3  |-  ( ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A )  i^i  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  B
) )  C_  ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A )
3 relco 5327 . . . 4  |-  Rel  ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A )
4 vex 2919 . . . . . . . . 9  |-  z  e. 
_V
5 vex 2919 . . . . . . . . 9  |-  y  e. 
_V
64, 5brcnv 5014 . . . . . . . 8  |-  ( z `' ( 1st  |`  ( _V  X.  _V ) ) y  <->  y ( 1st  |`  ( _V  X.  _V ) ) z )
74brres 5111 . . . . . . . . 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 1641 . . . . 5  |-  ( E. z ( x A z  /\  z `' ( 1st  |`  ( _V  X.  _V ) ) y )  ->  y  e.  ( _V  X.  _V ) )
12 vex 2919 . . . . . 6  |-  x  e. 
_V
1312, 5opelco 5003 . . . . 5  |-  ( <.
x ,  y >.  e.  ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A
)  <->  E. z ( x A z  /\  z `' ( 1st  |`  ( _V  X.  _V ) ) y ) )
14 opelxp 4867 . . . . . 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 4926 . . 3  |-  ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A )  C_  ( _V  X.  ( _V  X.  _V ) )
182, 17sstri 3317 . 2  |-  ( ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A )  i^i  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  B
) )  C_  ( _V  X.  ( _V  X.  _V ) )
191, 18eqsstri 3338 1  |-  ( A 
(x)  B )  C_  ( _V  X.  ( _V  X.  _V ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 359   E.wex 1547    e. wcel 1721   _Vcvv 2916    i^i cin 3279    C_ wss 3280   <.cop 3777   class class class wbr 4172    X. cxp 4835   `'ccnv 4836    |` cres 4839    o. ccom 4841   1stc1st 6306   2ndc2nd 6307    (x) ctxp 25587
This theorem is referenced by:  txprel  25633  brtxp2  25635  pprodss4v  25638
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-res 4849  df-txp 25609
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