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Theorem eloprabi 5850
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 2062 . . . . . 6  |-  ( w  =  A  ->  (
w  =  <. <. x ,  y >. ,  z
>. 
<->  A  =  <. <. x ,  y >. ,  z
>. ) )
21anbi1d 446 . . . . 5  |-  ( w  =  A  ->  (
( w  =  <. <.
x ,  y >. ,  z >.  /\  ph ) 
<->  ( A  =  <. <.
x ,  y >. ,  z >.  /\  ph ) ) )
323exbidv 1765 . . . 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 5544 . . . 4  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { w  |  E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) }
53, 4elab2g 2712 . . 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 169 . 2  |-  ( A  e.  { <. <. x ,  y >. ,  z
>.  |  ph }  ->  E. x E. y E. z ( A  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) )
7 vex 2577 . . . . . . . . . . . 12  |-  x  e. 
_V
8 vex 2577 . . . . . . . . . . . 12  |-  y  e. 
_V
97, 8opex 3994 . . . . . . . . . . 11  |-  <. x ,  y >.  e.  _V
10 vex 2577 . . . . . . . . . . 11  |-  z  e. 
_V
119, 10op1std 5803 . . . . . . . . . 10  |-  ( A  =  <. <. x ,  y
>. ,  z >.  -> 
( 1st `  A
)  =  <. x ,  y >. )
1211fveq2d 5210 . . . . . . . . 9  |-  ( A  =  <. <. x ,  y
>. ,  z >.  -> 
( 1st `  ( 1st `  A ) )  =  ( 1st `  <. x ,  y >. )
)
137, 8op1st 5801 . . . . . . . . 9  |-  ( 1st `  <. x ,  y
>. )  =  x
1412, 13syl6req 2105 . . . . . . . 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 5210 . . . . . . . . 9  |-  ( A  =  <. <. x ,  y
>. ,  z >.  -> 
( 2nd `  ( 1st `  A ) )  =  ( 2nd `  <. x ,  y >. )
)
187, 8op2nd 5802 . . . . . . . . 9  |-  ( 2nd `  <. x ,  y
>. )  =  y
1917, 18syl6req 2105 . . . . . . . 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 5804 . . . . . . . . 9  |-  ( A  =  <. <. x ,  y
>. ,  z >.  -> 
( 2nd `  A
)  =  z )
2322eqcomd 2061 . . . . . . . 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 207 . . . . . 6  |-  ( A  =  <. <. x ,  y
>. ,  z >.  -> 
( ph  <->  th ) )
2726biimpa 284 . . . . 5  |-  ( ( A  =  <. <. x ,  y >. ,  z
>.  /\  ph )  ->  th )
2827exlimiv 1505 . . . 4  |-  ( E. z ( A  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  th )
2928exlimiv 1505 . . 3  |-  ( E. y E. z ( A  =  <. <. x ,  y >. ,  z
>.  /\  ph )  ->  th )
3029exlimiv 1505 . 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 101    <-> wb 102    = wceq 1259   E.wex 1397    e. wcel 1409   <.cop 3406   ` cfv 4930   {coprab 5541   1stc1st 5793   2ndc2nd 5794
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-iota 4895  df-fun 4932  df-fv 4938  df-oprab 5544  df-1st 5795  df-2nd 5796
This theorem is referenced by: (None)
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