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Theorem eloprabi 6847
Description: A consequence of membership in an operation class abstraction, using ordered pair extractors. (Contributed by NM, 6-Nov-2006.) (Revised by David Abernethy, 19-Jun-2012.)
Hypotheses
Ref Expression
eloprabi.1  |-  ( x  =  ( 1st `  ( 1st `  A ) )  ->  ( ph  <->  ps )
)
eloprabi.2  |-  ( y  =  ( 2nd `  ( 1st `  A ) )  ->  ( ps  <->  ch )
)
eloprabi.3  |-  ( z  =  ( 2nd `  A
)  ->  ( ch  <->  th ) )
Assertion
Ref Expression
eloprabi  |-  ( A  e.  { <. <. x ,  y >. ,  z
>.  |  ph }  ->  th )
Distinct variable groups:    x, y,
z, A    th, x, y, z
Allowed substitution hints:    ph( x, y, z)    ps( x, y, z)    ch( x, y, z)

Proof of Theorem eloprabi
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2447 . . . . . 6  |-  ( w  =  A  ->  (
w  =  <. <. x ,  y >. ,  z
>. 
<->  A  =  <. <. x ,  y >. ,  z
>. ) )
21anbi1d 704 . . . . 5  |-  ( w  =  A  ->  (
( w  =  <. <.
x ,  y >. ,  z >.  /\  ph ) 
<->  ( A  =  <. <.
x ,  y >. ,  z >.  /\  ph ) ) )
323exbidv 1704 . . . 4  |-  ( w  =  A  ->  ( E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  <->  E. x E. y E. z ( A  =  <. <. x ,  y >. ,  z
>.  /\  ph ) ) )
4 df-oprab 6285 . . . 4  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { w  |  E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) }
53, 4elab2g 3234 . . 3  |-  ( A  e.  { <. <. x ,  y >. ,  z
>.  |  ph }  ->  ( A  e.  { <. <.
x ,  y >. ,  z >.  |  ph } 
<->  E. x E. y E. z ( A  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) ) )
65ibi 241 . 2  |-  ( A  e.  { <. <. x ,  y >. ,  z
>.  |  ph }  ->  E. x E. y E. z ( A  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) )
7 opex 4701 . . . . . . . . . . 11  |-  <. x ,  y >.  e.  _V
8 vex 3098 . . . . . . . . . . 11  |-  z  e. 
_V
97, 8op1std 6795 . . . . . . . . . 10  |-  ( A  =  <. <. x ,  y
>. ,  z >.  -> 
( 1st `  A
)  =  <. x ,  y >. )
109fveq2d 5860 . . . . . . . . 9  |-  ( A  =  <. <. x ,  y
>. ,  z >.  -> 
( 1st `  ( 1st `  A ) )  =  ( 1st `  <. x ,  y >. )
)
11 vex 3098 . . . . . . . . . 10  |-  x  e. 
_V
12 vex 3098 . . . . . . . . . 10  |-  y  e. 
_V
1311, 12op1st 6793 . . . . . . . . 9  |-  ( 1st `  <. x ,  y
>. )  =  x
1410, 13syl6req 2501 . . . . . . . 8  |-  ( A  =  <. <. x ,  y
>. ,  z >.  ->  x  =  ( 1st `  ( 1st `  A
) ) )
15 eloprabi.1 . . . . . . . 8  |-  ( x  =  ( 1st `  ( 1st `  A ) )  ->  ( ph  <->  ps )
)
1614, 15syl 16 . . . . . . 7  |-  ( A  =  <. <. x ,  y
>. ,  z >.  -> 
( ph  <->  ps ) )
179fveq2d 5860 . . . . . . . . 9  |-  ( A  =  <. <. x ,  y
>. ,  z >.  -> 
( 2nd `  ( 1st `  A ) )  =  ( 2nd `  <. x ,  y >. )
)
1811, 12op2nd 6794 . . . . . . . . 9  |-  ( 2nd `  <. x ,  y
>. )  =  y
1917, 18syl6req 2501 . . . . . . . 8  |-  ( A  =  <. <. x ,  y
>. ,  z >.  -> 
y  =  ( 2nd `  ( 1st `  A
) ) )
20 eloprabi.2 . . . . . . . 8  |-  ( y  =  ( 2nd `  ( 1st `  A ) )  ->  ( ps  <->  ch )
)
2119, 20syl 16 . . . . . . 7  |-  ( A  =  <. <. x ,  y
>. ,  z >.  -> 
( ps  <->  ch )
)
227, 8op2ndd 6796 . . . . . . . . 9  |-  ( A  =  <. <. x ,  y
>. ,  z >.  -> 
( 2nd `  A
)  =  z )
2322eqcomd 2451 . . . . . . . 8  |-  ( A  =  <. <. x ,  y
>. ,  z >.  -> 
z  =  ( 2nd `  A ) )
24 eloprabi.3 . . . . . . . 8  |-  ( z  =  ( 2nd `  A
)  ->  ( ch  <->  th ) )
2523, 24syl 16 . . . . . . 7  |-  ( A  =  <. <. x ,  y
>. ,  z >.  -> 
( ch  <->  th )
)
2616, 21, 253bitrd 279 . . . . . 6  |-  ( A  =  <. <. x ,  y
>. ,  z >.  -> 
( ph  <->  th ) )
2726biimpa 484 . . . . 5  |-  ( ( A  =  <. <. x ,  y >. ,  z
>.  /\  ph )  ->  th )
2827exlimiv 1709 . . . 4  |-  ( E. z ( A  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  th )
2928exlimiv 1709 . . 3  |-  ( E. y E. z ( A  =  <. <. x ,  y >. ,  z
>.  /\  ph )  ->  th )
3029exlimiv 1709 . 2  |-  ( E. x E. y E. z ( A  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  th )
316, 30syl 16 1  |-  ( A  e.  { <. <. x ,  y >. ,  z
>.  |  ph }  ->  th )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383   E.wex 1599    e. wcel 1804   <.cop 4020   ` cfv 5578   {coprab 6282   1stc1st 6783   2ndc2nd 6784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-iota 5541  df-fun 5580  df-fv 5586  df-oprab 6285  df-1st 6785  df-2nd 6786
This theorem is referenced by: (None)
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