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Theorem eloprabi 5958
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 2094 . . . . . 6  |-  ( w  =  A  ->  (
w  =  <. <. x ,  y >. ,  z
>. 
<->  A  =  <. <. x ,  y >. ,  z
>. ) )
21anbi1d 453 . . . . 5  |-  ( w  =  A  ->  (
( w  =  <. <.
x ,  y >. ,  z >.  /\  ph ) 
<->  ( A  =  <. <.
x ,  y >. ,  z >.  /\  ph ) ) )
323exbidv 1797 . . . 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 5648 . . . 4  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { w  |  E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) }
53, 4elab2g 2762 . . 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 174 . 2  |-  ( A  e.  { <. <. x ,  y >. ,  z
>.  |  ph }  ->  E. x E. y E. z ( A  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) )
7 vex 2622 . . . . . . . . . . . 12  |-  x  e. 
_V
8 vex 2622 . . . . . . . . . . . 12  |-  y  e. 
_V
97, 8opex 4054 . . . . . . . . . . 11  |-  <. x ,  y >.  e.  _V
10 vex 2622 . . . . . . . . . . 11  |-  z  e. 
_V
119, 10op1std 5911 . . . . . . . . . 10  |-  ( A  =  <. <. x ,  y
>. ,  z >.  -> 
( 1st `  A
)  =  <. x ,  y >. )
1211fveq2d 5303 . . . . . . . . 9  |-  ( A  =  <. <. x ,  y
>. ,  z >.  -> 
( 1st `  ( 1st `  A ) )  =  ( 1st `  <. x ,  y >. )
)
137, 8op1st 5909 . . . . . . . . 9  |-  ( 1st `  <. x ,  y
>. )  =  x
1412, 13syl6req 2137 . . . . . . . 8  |-  ( A  =  <. <. x ,  y
>. ,  z >.  ->  x  =  ( 1st `  ( 1st `  A
) ) )
15 eloprabi.1 . . . . . . . 8  |-  ( x  =  ( 1st `  ( 1st `  A ) )  ->  ( ph  <->  ps )
)
1614, 15syl 14 . . . . . . 7  |-  ( A  =  <. <. x ,  y
>. ,  z >.  -> 
( ph  <->  ps ) )
1711fveq2d 5303 . . . . . . . . 9  |-  ( A  =  <. <. x ,  y
>. ,  z >.  -> 
( 2nd `  ( 1st `  A ) )  =  ( 2nd `  <. x ,  y >. )
)
187, 8op2nd 5910 . . . . . . . . 9  |-  ( 2nd `  <. x ,  y
>. )  =  y
1917, 18syl6req 2137 . . . . . . . 8  |-  ( A  =  <. <. x ,  y
>. ,  z >.  -> 
y  =  ( 2nd `  ( 1st `  A
) ) )
20 eloprabi.2 . . . . . . . 8  |-  ( y  =  ( 2nd `  ( 1st `  A ) )  ->  ( ps  <->  ch )
)
2119, 20syl 14 . . . . . . 7  |-  ( A  =  <. <. x ,  y
>. ,  z >.  -> 
( ps  <->  ch )
)
229, 10op2ndd 5912 . . . . . . . . 9  |-  ( A  =  <. <. x ,  y
>. ,  z >.  -> 
( 2nd `  A
)  =  z )
2322eqcomd 2093 . . . . . . . 8  |-  ( A  =  <. <. x ,  y
>. ,  z >.  -> 
z  =  ( 2nd `  A ) )
24 eloprabi.3 . . . . . . . 8  |-  ( z  =  ( 2nd `  A
)  ->  ( ch  <->  th ) )
2523, 24syl 14 . . . . . . 7  |-  ( A  =  <. <. x ,  y
>. ,  z >.  -> 
( ch  <->  th )
)
2616, 21, 253bitrd 212 . . . . . 6  |-  ( A  =  <. <. x ,  y
>. ,  z >.  -> 
( ph  <->  th ) )
2726biimpa 290 . . . . 5  |-  ( ( A  =  <. <. x ,  y >. ,  z
>.  /\  ph )  ->  th )
2827exlimiv 1534 . . . 4  |-  ( E. z ( A  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  th )
2928exlimiv 1534 . . 3  |-  ( E. y E. z ( A  =  <. <. x ,  y >. ,  z
>.  /\  ph )  ->  th )
3029exlimiv 1534 . 2  |-  ( E. x E. y E. z ( A  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  th )
316, 30syl 14 1  |-  ( A  e.  { <. <. x ,  y >. ,  z
>.  |  ph }  ->  th )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289   E.wex 1426    e. wcel 1438   <.cop 3447   ` cfv 5010   {coprab 5645   1stc1st 5901   2ndc2nd 5902
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3955  ax-pow 4007  ax-pr 4034  ax-un 4258
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-br 3844  df-opab 3898  df-mpt 3899  df-id 4118  df-xp 4442  df-rel 4443  df-cnv 4444  df-co 4445  df-dm 4446  df-rn 4447  df-iota 4975  df-fun 5012  df-fv 5018  df-oprab 5648  df-1st 5903  df-2nd 5904
This theorem is referenced by: (None)
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