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Theorem oprabex3 6191
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 4993 . 2  |-  ( H  X.  H )  e. 
_V
3 moeq 3112 . . . . . 6  |-  E* z 
z  =  R
43mosubop 4458 . . . . 5  |-  E* z E. u E. f ( y  =  <. u ,  f >.  /\  z  =  R )
54mosubop 4458 . . . 4  |-  E* z E. w E. v ( x  =  <. w ,  v >.  /\  E. u E. f ( y  =  <. u ,  f
>.  /\  z  =  R ) )
6 anass 632 . . . . . . . 8  |-  ( ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f
>. )  /\  z  =  R )  <->  ( x  =  <. w ,  v
>.  /\  ( y  = 
<. u ,  f >.  /\  z  =  R
) ) )
762exbii 1594 . . . . . . 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 1931 . . . . . . 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 242 . . . . . 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 1594 . . . . 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 2319 . . . 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 202 . . 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 6190 1  |-  F  e. 
_V
Colors of variables: wff set class
Syntax hints:    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1726   E*wmo 2284   _Vcvv 2958   <.cop 3819    X. cxp 4879   {coprab 6085
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-oprab 6088
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