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Theorem eloprabi 6202
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 2302 . . . . . 6  |-  ( w  =  A  ->  (
w  =  <. <. x ,  y >. ,  z
>. 
<->  A  =  <. <. x ,  y >. ,  z
>. ) )
21anbi1d 685 . . . . 5  |-  ( w  =  A  ->  (
( w  =  <. <.
x ,  y >. ,  z >.  /\  ph ) 
<->  ( A  =  <. <.
x ,  y >. ,  z >.  /\  ph ) ) )
323exbidv 1619 . . . 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 5878 . . . 4  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { w  |  E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) }
53, 4elab2g 2929 . . 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 232 . 2  |-  ( A  e.  { <. <. x ,  y >. ,  z
>.  |  ph }  ->  E. x E. y E. z ( A  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) )
7 opex 4253 . . . . . . . . . . 11  |-  <. x ,  y >.  e.  _V
8 vex 2804 . . . . . . . . . . 11  |-  z  e. 
_V
97, 8op1std 6146 . . . . . . . . . 10  |-  ( A  =  <. <. x ,  y
>. ,  z >.  -> 
( 1st `  A
)  =  <. x ,  y >. )
109fveq2d 5545 . . . . . . . . 9  |-  ( A  =  <. <. x ,  y
>. ,  z >.  -> 
( 1st `  ( 1st `  A ) )  =  ( 1st `  <. x ,  y >. )
)
11 vex 2804 . . . . . . . . . 10  |-  x  e. 
_V
12 vex 2804 . . . . . . . . . 10  |-  y  e. 
_V
1311, 12op1st 6144 . . . . . . . . 9  |-  ( 1st `  <. x ,  y
>. )  =  x
1410, 13syl6req 2345 . . . . . . . 8  |-  ( A  =  <. <. x ,  y
>. ,  z >.  ->  x  =  ( 1st `  ( 1st `  A
) ) )
15 eloprabi.1 . . . . . . . 8  |-  ( x  =  ( 1st `  ( 1st `  A ) )  ->  ( ph  <->  ps )
)
1614, 15syl 15 . . . . . . 7  |-  ( A  =  <. <. x ,  y
>. ,  z >.  -> 
( ph  <->  ps ) )
179fveq2d 5545 . . . . . . . . 9  |-  ( A  =  <. <. x ,  y
>. ,  z >.  -> 
( 2nd `  ( 1st `  A ) )  =  ( 2nd `  <. x ,  y >. )
)
1811, 12op2nd 6145 . . . . . . . . 9  |-  ( 2nd `  <. x ,  y
>. )  =  y
1917, 18syl6req 2345 . . . . . . . 8  |-  ( A  =  <. <. x ,  y
>. ,  z >.  -> 
y  =  ( 2nd `  ( 1st `  A
) ) )
20 eloprabi.2 . . . . . . . 8  |-  ( y  =  ( 2nd `  ( 1st `  A ) )  ->  ( ps  <->  ch )
)
2119, 20syl 15 . . . . . . 7  |-  ( A  =  <. <. x ,  y
>. ,  z >.  -> 
( ps  <->  ch )
)
227, 8op2ndd 6147 . . . . . . . . 9  |-  ( A  =  <. <. x ,  y
>. ,  z >.  -> 
( 2nd `  A
)  =  z )
2322eqcomd 2301 . . . . . . . 8  |-  ( A  =  <. <. x ,  y
>. ,  z >.  -> 
z  =  ( 2nd `  A ) )
24 eloprabi.3 . . . . . . . 8  |-  ( z  =  ( 2nd `  A
)  ->  ( ch  <->  th ) )
2523, 24syl 15 . . . . . . 7  |-  ( A  =  <. <. x ,  y
>. ,  z >.  -> 
( ch  <->  th )
)
2616, 21, 253bitrd 270 . . . . . 6  |-  ( A  =  <. <. x ,  y
>. ,  z >.  -> 
( ph  <->  th ) )
2726biimpa 470 . . . . 5  |-  ( ( A  =  <. <. x ,  y >. ,  z
>.  /\  ph )  ->  th )
2827exlimiv 1624 . . . 4  |-  ( E. z ( A  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  th )
2928exlimiv 1624 . . 3  |-  ( E. y E. z ( A  =  <. <. x ,  y >. ,  z
>.  /\  ph )  ->  th )
3029exlimiv 1624 . 2  |-  ( E. x E. y E. z ( A  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  th )
316, 30syl 15 1  |-  ( A  e.  { <. <. x ,  y >. ,  z
>.  |  ph }  ->  th )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   <.cop 3656   ` cfv 5271   {coprab 5875   1stc1st 6136   2ndc2nd 6137
This theorem is referenced by:  prismorcset2  26021  domcatfun  26028  codcatfun  26037
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-oprab 5878  df-1st 6138  df-2nd 6139
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