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Theorem oprabex3 6155
Description: Existence of an operation class abstraction (special case). (Contributed by NM, 19-Oct-2004.)
Hypotheses
Ref Expression
oprabex3.1  |-  H  e. 
_V
oprabex3.2  |-  F  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  ( H  X.  H
)  /\  y  e.  ( H  X.  H
) )  /\  E. w E. v E. u E. f ( ( x  =  <. w ,  v
>.  /\  y  =  <. u ,  f >. )  /\  z  =  R
) ) }
Assertion
Ref Expression
oprabex3  |-  F  e. 
_V
Distinct variable groups:    x, y,
z, w, v, u, f, H    x, R, y, z
Allowed substitution hints:    R( w, v, u, f)    F( x, y, z, w, v, u, f)

Proof of Theorem oprabex3
StepHypRef Expression
1 oprabex3.1 . . 3  |-  H  e. 
_V
21, 1xpex 4957 . 2  |-  ( H  X.  H )  e. 
_V
3 moeq 3078 . . . . . 6  |-  E* z 
z  =  R
43mosubop 4423 . . . . 5  |-  E* z E. u E. f ( y  =  <. u ,  f >.  /\  z  =  R )
54mosubop 4423 . . . 4  |-  E* z E. w E. v ( x  =  <. w ,  v >.  /\  E. u E. f ( y  =  <. u ,  f
>.  /\  z  =  R ) )
6 anass 631 . . . . . . . 8  |-  ( ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f
>. )  /\  z  =  R )  <->  ( x  =  <. w ,  v
>.  /\  ( y  = 
<. u ,  f >.  /\  z  =  R
) ) )
762exbii 1590 . . . . . . 7  |-  ( E. u E. f ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f
>. )  /\  z  =  R )  <->  E. u E. f ( x  = 
<. w ,  v >.  /\  ( y  =  <. u ,  f >.  /\  z  =  R ) ) )
8 19.42vv 1926 . . . . . . 7  |-  ( E. u E. f ( x  =  <. w ,  v >.  /\  (
y  =  <. u ,  f >.  /\  z  =  R ) )  <->  ( x  =  <. w ,  v
>.  /\  E. u E. f ( y  = 
<. u ,  f >.  /\  z  =  R
) ) )
97, 8bitri 241 . . . . . 6  |-  ( E. u E. f ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f
>. )  /\  z  =  R )  <->  ( x  =  <. w ,  v
>.  /\  E. u E. f ( y  = 
<. u ,  f >.  /\  z  =  R
) ) )
1092exbii 1590 . . . . 5  |-  ( E. w E. v E. u E. f ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f
>. )  /\  z  =  R )  <->  E. w E. v ( x  = 
<. w ,  v >.  /\  E. u E. f
( y  =  <. u ,  f >.  /\  z  =  R ) ) )
1110mobii 2298 . . . 4  |-  ( E* z E. w E. v E. u E. f
( ( x  = 
<. w ,  v >.  /\  y  =  <. u ,  f >. )  /\  z  =  R
)  <->  E* z E. w E. v ( x  = 
<. w ,  v >.  /\  E. u E. f
( y  =  <. u ,  f >.  /\  z  =  R ) ) )
125, 11mpbir 201 . . 3  |-  E* z E. w E. v E. u E. f ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f
>. )  /\  z  =  R )
1312a1i 11 . 2  |-  ( ( x  e.  ( H  X.  H )  /\  y  e.  ( H  X.  H ) )  ->  E* z E. w E. v E. u E. f
( ( x  = 
<. w ,  v >.  /\  y  =  <. u ,  f >. )  /\  z  =  R
) )
14 oprabex3.2 . 2  |-  F  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  ( H  X.  H
)  /\  y  e.  ( H  X.  H
) )  /\  E. w E. v E. u E. f ( ( x  =  <. w ,  v
>.  /\  y  =  <. u ,  f >. )  /\  z  =  R
) ) }
152, 2, 13, 14oprabex 6154 1  |-  F  e. 
_V
Colors of variables: wff set class
Syntax hints:    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1721   E*wmo 2263   _Vcvv 2924   <.cop 3785    X. cxp 4843   {coprab 6049
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-oprab 6052
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