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Theorem txpss3v 24976
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 24953 . 2  |-  ( A 
(x)  B )  =  ( ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A )  i^i  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  B ) )
2 inss1 3465 . . 3  |-  ( ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A )  i^i  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  B
) )  C_  ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A )
3 relco 5250 . . . 4  |-  Rel  ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A )
4 vex 2867 . . . . . . . . 9  |-  z  e. 
_V
5 vex 2867 . . . . . . . . 9  |-  y  e. 
_V
64, 5brcnv 4943 . . . . . . . 8  |-  ( z `' ( 1st  |`  ( _V  X.  _V ) ) y  <->  y ( 1st  |`  ( _V  X.  _V ) ) z )
74brres 5040 . . . . . . . . 9  |-  ( y ( 1st  |`  ( _V  X.  _V ) ) z  <->  ( y 1st z  /\  y  e.  ( _V  X.  _V ) ) )
87simprbi 450 . . . . . . . 8  |-  ( y ( 1st  |`  ( _V  X.  _V ) ) z  ->  y  e.  ( _V  X.  _V )
)
96, 8sylbi 187 . . . . . . 7  |-  ( z `' ( 1st  |`  ( _V  X.  _V ) ) y  ->  y  e.  ( _V  X.  _V )
)
109adantl 452 . . . . . 6  |-  ( ( x A z  /\  z `' ( 1st  |`  ( _V  X.  _V ) ) y )  ->  y  e.  ( _V  X.  _V ) )
1110exlimiv 1634 . . . . 5  |-  ( E. z ( x A z  /\  z `' ( 1st  |`  ( _V  X.  _V ) ) y )  ->  y  e.  ( _V  X.  _V ) )
12 vex 2867 . . . . . 6  |-  x  e. 
_V
1312, 5opelco 4932 . . . . 5  |-  ( <.
x ,  y >.  e.  ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A
)  <->  E. z ( x A z  /\  z `' ( 1st  |`  ( _V  X.  _V ) ) y ) )
14 opelxp 4798 . . . . . 6  |-  ( <.
x ,  y >.  e.  ( _V  X.  ( _V  X.  _V ) )  <-> 
( x  e.  _V  /\  y  e.  ( _V 
X.  _V ) ) )
1512, 14mpbiran 884 . . . . 5  |-  ( <.
x ,  y >.  e.  ( _V  X.  ( _V  X.  _V ) )  <-> 
y  e.  ( _V 
X.  _V ) )
1611, 13, 153imtr4i 257 . . . 4  |-  ( <.
x ,  y >.  e.  ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A
)  ->  <. x ,  y >.  e.  ( _V  X.  ( _V  X.  _V ) ) )
173, 16relssi 4857 . . 3  |-  ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A )  C_  ( _V  X.  ( _V  X.  _V ) )
182, 17sstri 3264 . 2  |-  ( ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  A )  i^i  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  B
) )  C_  ( _V  X.  ( _V  X.  _V ) )
191, 18eqsstri 3284 1  |-  ( A 
(x)  B )  C_  ( _V  X.  ( _V  X.  _V ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1541    e. wcel 1710   _Vcvv 2864    i^i cin 3227    C_ wss 3228   <.cop 3719   class class class wbr 4102    X. cxp 4766   `'ccnv 4767    |` cres 4770    o. ccom 4772   1stc1st 6204   2ndc2nd 6205    (x) ctxp 24931
This theorem is referenced by:  txprel  24977  brtxp2  24979  pprodss4v  24982
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pr 4293
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-br 4103  df-opab 4157  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-res 4780  df-txp 24953
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