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Theorem eloprabi 6173
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 2177 . . . . . 6  |-  ( w  =  A  ->  (
w  =  <. <. x ,  y >. ,  z
>. 
<->  A  =  <. <. x ,  y >. ,  z
>. ) )
21anbi1d 462 . . . . 5  |-  ( w  =  A  ->  (
( w  =  <. <.
x ,  y >. ,  z >.  /\  ph ) 
<->  ( A  =  <. <.
x ,  y >. ,  z >.  /\  ph ) ) )
323exbidv 1862 . . . 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 5855 . . . 4  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { w  |  E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) }
53, 4elab2g 2877 . . 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 175 . 2  |-  ( A  e.  { <. <. x ,  y >. ,  z
>.  |  ph }  ->  E. x E. y E. z ( A  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) )
7 vex 2733 . . . . . . . . . . . 12  |-  x  e. 
_V
8 vex 2733 . . . . . . . . . . . 12  |-  y  e. 
_V
97, 8opex 4212 . . . . . . . . . . 11  |-  <. x ,  y >.  e.  _V
10 vex 2733 . . . . . . . . . . 11  |-  z  e. 
_V
119, 10op1std 6125 . . . . . . . . . 10  |-  ( A  =  <. <. x ,  y
>. ,  z >.  -> 
( 1st `  A
)  =  <. x ,  y >. )
1211fveq2d 5498 . . . . . . . . 9  |-  ( A  =  <. <. x ,  y
>. ,  z >.  -> 
( 1st `  ( 1st `  A ) )  =  ( 1st `  <. x ,  y >. )
)
137, 8op1st 6123 . . . . . . . . 9  |-  ( 1st `  <. x ,  y
>. )  =  x
1412, 13eqtr2di 2220 . . . . . . . 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 5498 . . . . . . . . 9  |-  ( A  =  <. <. x ,  y
>. ,  z >.  -> 
( 2nd `  ( 1st `  A ) )  =  ( 2nd `  <. x ,  y >. )
)
187, 8op2nd 6124 . . . . . . . . 9  |-  ( 2nd `  <. x ,  y
>. )  =  y
1917, 18eqtr2di 2220 . . . . . . . 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 6126 . . . . . . . . 9  |-  ( A  =  <. <. x ,  y
>. ,  z >.  -> 
( 2nd `  A
)  =  z )
2322eqcomd 2176 . . . . . . . 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 213 . . . . . 6  |-  ( A  =  <. <. x ,  y
>. ,  z >.  -> 
( ph  <->  th ) )
2726biimpa 294 . . . . 5  |-  ( ( A  =  <. <. x ,  y >. ,  z
>.  /\  ph )  ->  th )
2827exlimiv 1591 . . . 4  |-  ( E. z ( A  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  th )
2928exlimiv 1591 . . 3  |-  ( E. y E. z ( A  =  <. <. x ,  y >. ,  z
>.  /\  ph )  ->  th )
3029exlimiv 1591 . 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 103    <-> wb 104    = wceq 1348   E.wex 1485    e. wcel 2141   <.cop 3584   ` cfv 5196   {coprab 5852   1stc1st 6115   2ndc2nd 6116
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-iota 5158  df-fun 5198  df-fv 5204  df-oprab 5855  df-1st 6117  df-2nd 6118
This theorem is referenced by: (None)
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