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Theorem eloi 25189
Description: A consequence of membership in a class abstraction whose elements belong to  ( ( _V 
X.  _V )  X.  ( _V  X.  _V ) ) using ordered pair extractors. (Used by category theory). (Contributed by FL, 24-Sep-2007.)
Hypotheses
Ref Expression
eloi.1  |-  ( y  =  ( 1st `  ( 1st `  A ) )  ->  ( ph  <->  ps )
)
eloi.2  |-  ( z  =  ( 2nd `  ( 1st `  A ) )  ->  ( ps  <->  ch )
)
eloi.3  |-  ( v  =  ( 1st `  ( 2nd `  A ) )  ->  ( ch  <->  th )
)
eloi.4  |-  ( w  =  ( 2nd `  ( 2nd `  A ) )  ->  ( th  <->  ta )
)
Assertion
Ref Expression
eloi  |-  ( A  e.  { x  |  E. y E. z E. v E. w ( x  =  <. <. y ,  z >. ,  <. v ,  w >. >.  /\  ph ) }  ->  ta )
Distinct variable groups:    v, A, w, x, y, z    ph, x    ta, v, w, y, z
Allowed substitution hints:    ph( y, z, w, v)    ps( x, y, z, w, v)    ch( x, y, z, w, v)    th( x, y, z, w, v)    ta( x)

Proof of Theorem eloi
StepHypRef Expression
1 eqeq1 2302 . . . . . . 7  |-  ( x  =  A  ->  (
x  =  <. <. y ,  z >. ,  <. v ,  w >. >.  <->  A  =  <. <. y ,  z
>. ,  <. v ,  w >. >. ) )
21anbi1d 685 . . . . . 6  |-  ( x  =  A  ->  (
( x  =  <. <.
y ,  z >. ,  <. v ,  w >. >.  /\  ph )  <->  ( A  =  <. <. y ,  z
>. ,  <. v ,  w >. >.  /\  ph ) ) )
324exbidv 1620 . . . . 5  |-  ( x  =  A  ->  ( E. y E. z E. v E. w ( x  =  <. <. y ,  z >. ,  <. v ,  w >. >.  /\  ph ) 
<->  E. y E. z E. v E. w ( A  =  <. <. y ,  z >. ,  <. v ,  w >. >.  /\  ph ) ) )
43elabg 2928 . . . 4  |-  ( A  e.  { x  |  E. y E. z E. v E. w ( x  =  <. <. y ,  z >. ,  <. v ,  w >. >.  /\  ph ) }  ->  ( A  e.  { x  |  E. y E. z E. v E. w ( x  =  <. <. y ,  z >. ,  <. v ,  w >. >.  /\  ph ) }  <->  E. y E. z E. v E. w ( A  =  <. <. y ,  z >. ,  <. v ,  w >. >.  /\  ph ) ) )
54ibi 232 . . 3  |-  ( A  e.  { x  |  E. y E. z E. v E. w ( x  =  <. <. y ,  z >. ,  <. v ,  w >. >.  /\  ph ) }  ->  E. y E. z E. v E. w ( A  = 
<. <. y ,  z
>. ,  <. v ,  w >. >.  /\  ph ) )
6 id 19 . . . . . . . . 9  |-  ( A  =  <. <. y ,  z
>. ,  <. v ,  w >. >.  ->  A  =  <. <. y ,  z
>. ,  <. v ,  w >. >. )
7 opex 4253 . . . . . . . . . 10  |-  <. y ,  z >.  e.  _V
8 opex 4253 . . . . . . . . . 10  |-  <. v ,  w >.  e.  _V
97, 8opelvv 4751 . . . . . . . . 9  |-  <. <. y ,  z >. ,  <. v ,  w >. >.  e.  ( _V  X.  _V )
106, 9syl6eqel 2384 . . . . . . . 8  |-  ( A  =  <. <. y ,  z
>. ,  <. v ,  w >. >.  ->  A  e.  ( _V  X.  _V )
)
11 1st2nd2 6175 . . . . . . . 8  |-  ( A  e.  ( _V  X.  _V )  ->  A  = 
<. ( 1st `  A
) ,  ( 2nd `  A ) >. )
1210, 11syl 15 . . . . . . 7  |-  ( A  =  <. <. y ,  z
>. ,  <. v ,  w >. >.  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
137, 8op1std 6146 . . . . . . . . . 10  |-  ( A  =  <. <. y ,  z
>. ,  <. v ,  w >. >.  ->  ( 1st `  A )  =  <. y ,  z >. )
14 vex 2804 . . . . . . . . . . 11  |-  y  e. 
_V
15 vex 2804 . . . . . . . . . . 11  |-  z  e. 
_V
1614, 15opelvv 4751 . . . . . . . . . 10  |-  <. y ,  z >.  e.  ( _V  X.  _V )
1713, 16syl6eqel 2384 . . . . . . . . 9  |-  ( A  =  <. <. y ,  z
>. ,  <. v ,  w >. >.  ->  ( 1st `  A )  e.  ( _V  X.  _V )
)
18 1st2nd2 6175 . . . . . . . . 9  |-  ( ( 1st `  A )  e.  ( _V  X.  _V )  ->  ( 1st `  A )  =  <. ( 1st `  ( 1st `  A ) ) ,  ( 2nd `  ( 1st `  A ) )
>. )
1917, 18syl 15 . . . . . . . 8  |-  ( A  =  <. <. y ,  z
>. ,  <. v ,  w >. >.  ->  ( 1st `  A )  =  <. ( 1st `  ( 1st `  A ) ) ,  ( 2nd `  ( 1st `  A ) )
>. )
207, 8op2ndd 6147 . . . . . . . . . 10  |-  ( A  =  <. <. y ,  z
>. ,  <. v ,  w >. >.  ->  ( 2nd `  A )  =  <. v ,  w >. )
21 vex 2804 . . . . . . . . . . 11  |-  v  e. 
_V
22 vex 2804 . . . . . . . . . . 11  |-  w  e. 
_V
2321, 22opelvv 4751 . . . . . . . . . 10  |-  <. v ,  w >.  e.  ( _V  X.  _V )
2420, 23syl6eqel 2384 . . . . . . . . 9  |-  ( A  =  <. <. y ,  z
>. ,  <. v ,  w >. >.  ->  ( 2nd `  A )  e.  ( _V  X.  _V )
)
25 1st2nd2 6175 . . . . . . . . 9  |-  ( ( 2nd `  A )  e.  ( _V  X.  _V )  ->  ( 2nd `  A )  =  <. ( 1st `  ( 2nd `  A ) ) ,  ( 2nd `  ( 2nd `  A ) )
>. )
2624, 25syl 15 . . . . . . . 8  |-  ( A  =  <. <. y ,  z
>. ,  <. v ,  w >. >.  ->  ( 2nd `  A )  =  <. ( 1st `  ( 2nd `  A ) ) ,  ( 2nd `  ( 2nd `  A ) )
>. )
2719, 26opeq12d 3820 . . . . . . 7  |-  ( A  =  <. <. y ,  z
>. ,  <. v ,  w >. >.  ->  <. ( 1st `  A ) ,  ( 2nd `  A )
>.  =  <. <. ( 1st `  ( 1st `  A
) ) ,  ( 2nd `  ( 1st `  A ) ) >. ,  <. ( 1st `  ( 2nd `  A ) ) ,  ( 2nd `  ( 2nd `  A ) )
>. >. )
2812, 27eqtrd 2328 . . . . . 6  |-  ( A  =  <. <. y ,  z
>. ,  <. v ,  w >. >.  ->  A  =  <. <. ( 1st `  ( 1st `  A ) ) ,  ( 2nd `  ( 1st `  A ) )
>. ,  <. ( 1st `  ( 2nd `  A
) ) ,  ( 2nd `  ( 2nd `  A ) ) >. >. )
2928adantr 451 . . . . 5  |-  ( ( A  =  <. <. y ,  z >. ,  <. v ,  w >. >.  /\  ph )  ->  A  =  <. <.
( 1st `  ( 1st `  A ) ) ,  ( 2nd `  ( 1st `  A ) )
>. ,  <. ( 1st `  ( 2nd `  A
) ) ,  ( 2nd `  ( 2nd `  A ) ) >. >. )
3029exlimivv 1625 . . . 4  |-  ( E. v E. w ( A  =  <. <. y ,  z >. ,  <. v ,  w >. >.  /\  ph )  ->  A  =  <. <.
( 1st `  ( 1st `  A ) ) ,  ( 2nd `  ( 1st `  A ) )
>. ,  <. ( 1st `  ( 2nd `  A
) ) ,  ( 2nd `  ( 2nd `  A ) ) >. >. )
3130exlimivv 1625 . . 3  |-  ( E. y E. z E. v E. w ( A  =  <. <. y ,  z >. ,  <. v ,  w >. >.  /\  ph )  ->  A  =  <. <.
( 1st `  ( 1st `  A ) ) ,  ( 2nd `  ( 1st `  A ) )
>. ,  <. ( 1st `  ( 2nd `  A
) ) ,  ( 2nd `  ( 2nd `  A ) ) >. >. )
325, 31syl 15 . 2  |-  ( A  e.  { x  |  E. y E. z E. v E. w ( x  =  <. <. y ,  z >. ,  <. v ,  w >. >.  /\  ph ) }  ->  A  = 
<. <. ( 1st `  ( 1st `  A ) ) ,  ( 2nd `  ( 1st `  A ) )
>. ,  <. ( 1st `  ( 2nd `  A
) ) ,  ( 2nd `  ( 2nd `  A ) ) >. >. )
33 eleq1 2356 . . . 4  |-  ( A  =  <. <. ( 1st `  ( 1st `  A ) ) ,  ( 2nd `  ( 1st `  A ) )
>. ,  <. ( 1st `  ( 2nd `  A
) ) ,  ( 2nd `  ( 2nd `  A ) ) >. >.  ->  ( A  e. 
{ x  |  E. y E. z E. v E. w ( x  = 
<. <. y ,  z
>. ,  <. v ,  w >. >.  /\  ph ) }  <->  <. <. ( 1st `  ( 1st `  A ) ) ,  ( 2nd `  ( 1st `  A ) )
>. ,  <. ( 1st `  ( 2nd `  A
) ) ,  ( 2nd `  ( 2nd `  A ) ) >. >.  e.  { x  |  E. y E. z E. v E. w ( x  =  <. <. y ,  z >. ,  <. v ,  w >. >.  /\  ph ) } ) )
34 fvex 5555 . . . . . 6  |-  ( 1st `  ( 1st `  A
) )  e.  _V
35 fvex 5555 . . . . . 6  |-  ( 2nd `  ( 1st `  A
) )  e.  _V
36 fvex 5555 . . . . . 6  |-  ( 1st `  ( 2nd `  A
) )  e.  _V
3734, 35, 363pm3.2i 1130 . . . . 5  |-  ( ( 1st `  ( 1st `  A ) )  e. 
_V  /\  ( 2nd `  ( 1st `  A
) )  e.  _V  /\  ( 1st `  ( 2nd `  A ) )  e.  _V )
38 fvex 5555 . . . . 5  |-  ( 2nd `  ( 2nd `  A
) )  e.  _V
39 eloi.1 . . . . . 6  |-  ( y  =  ( 1st `  ( 1st `  A ) )  ->  ( ph  <->  ps )
)
40 eloi.2 . . . . . 6  |-  ( z  =  ( 2nd `  ( 1st `  A ) )  ->  ( ps  <->  ch )
)
41 eloi.3 . . . . . 6  |-  ( v  =  ( 1st `  ( 2nd `  A ) )  ->  ( ch  <->  th )
)
42 eloi.4 . . . . . 6  |-  ( w  =  ( 2nd `  ( 2nd `  A ) )  ->  ( th  <->  ta )
)
4339, 40, 41, 42elo 25144 . . . . 5  |-  ( ( ( ( 1st `  ( 1st `  A ) )  e.  _V  /\  ( 2nd `  ( 1st `  A
) )  e.  _V  /\  ( 1st `  ( 2nd `  A ) )  e.  _V )  /\  ( 2nd `  ( 2nd `  A ) )  e. 
_V )  ->  ( <. <. ( 1st `  ( 1st `  A ) ) ,  ( 2nd `  ( 1st `  A ) )
>. ,  <. ( 1st `  ( 2nd `  A
) ) ,  ( 2nd `  ( 2nd `  A ) ) >. >.  e.  { x  |  E. y E. z E. v E. w ( x  =  <. <. y ,  z >. ,  <. v ,  w >. >.  /\  ph ) }  <->  ta ) )
4437, 38, 43mp2an 653 . . . 4  |-  ( <. <. ( 1st `  ( 1st `  A ) ) ,  ( 2nd `  ( 1st `  A ) )
>. ,  <. ( 1st `  ( 2nd `  A
) ) ,  ( 2nd `  ( 2nd `  A ) ) >. >.  e.  { x  |  E. y E. z E. v E. w ( x  =  <. <. y ,  z >. ,  <. v ,  w >. >.  /\  ph ) }  <->  ta )
4533, 44syl6bb 252 . . 3  |-  ( A  =  <. <. ( 1st `  ( 1st `  A ) ) ,  ( 2nd `  ( 1st `  A ) )
>. ,  <. ( 1st `  ( 2nd `  A
) ) ,  ( 2nd `  ( 2nd `  A ) ) >. >.  ->  ( A  e. 
{ x  |  E. y E. z E. v E. w ( x  = 
<. <. y ,  z
>. ,  <. v ,  w >. >.  /\  ph ) }  <->  ta ) )
4645biimpcd 215 . 2  |-  ( A  e.  { x  |  E. y E. z E. v E. w ( x  =  <. <. y ,  z >. ,  <. v ,  w >. >.  /\  ph ) }  ->  ( A  =  <. <. ( 1st `  ( 1st `  A ) ) ,  ( 2nd `  ( 1st `  A ) )
>. ,  <. ( 1st `  ( 2nd `  A
) ) ,  ( 2nd `  ( 2nd `  A ) ) >. >.  ->  ta ) )
4732, 46mpd 14 1  |-  ( A  e.  { x  |  E. y E. z E. v E. w ( x  =  <. <. y ,  z >. ,  <. v ,  w >. >.  /\  ph ) }  ->  ta )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1531    = wceq 1632    e. wcel 1696   {cab 2282   _Vcvv 2801   <.cop 3656    X. cxp 4703   ` cfv 5271   1stc1st 6136   2ndc2nd 6137
This theorem is referenced by:  vecval3b  25555  algi  25830  dedi  25840  cati  25858
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fv 5279  df-1st 6138  df-2nd 6139
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