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Theorem ovi3 5763
Description: The value of an operation class abstraction. Special case. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 29-Dec-2014.)
Hypotheses
Ref Expression
ovi3.1  |-  ( ( ( A  e.  H  /\  B  e.  H
)  /\  ( C  e.  H  /\  D  e.  H ) )  ->  S  e.  ( H  X.  H ) )
ovi3.2  |-  ( ( ( w  =  A  /\  v  =  B )  /\  ( u  =  C  /\  f  =  D ) )  ->  R  =  S )
ovi3.3  |-  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
ovi3  |-  ( ( ( A  e.  H  /\  B  e.  H
)  /\  ( C  e.  H  /\  D  e.  H ) )  -> 
( <. A ,  B >. F <. C ,  D >. )  =  S )
Distinct variable groups:    u, f, v, w, x, y, z, A    B, f, u, v, w, x, y, z   
x, R, y, z    C, f, u, v, w, y, z    D, f, u, v, w, y, z    f, H, u, v, w, x, y, z    S, f, u, v, w, z
Allowed substitution hints:    C( x)    D( x)    R( w, v, u, f)    S( x, y)    F( x, y, z, w, v, u, f)

Proof of Theorem ovi3
StepHypRef Expression
1 ovi3.1 . . . 4  |-  ( ( ( A  e.  H  /\  B  e.  H
)  /\  ( C  e.  H  /\  D  e.  H ) )  ->  S  e.  ( H  X.  H ) )
2 elex 2630 . . . 4  |-  ( S  e.  ( H  X.  H )  ->  S  e.  _V )
31, 2syl 14 . . 3  |-  ( ( ( A  e.  H  /\  B  e.  H
)  /\  ( C  e.  H  /\  D  e.  H ) )  ->  S  e.  _V )
4 isset 2625 . . 3  |-  ( S  e.  _V  <->  E. z 
z  =  S )
53, 4sylib 120 . 2  |-  ( ( ( A  e.  H  /\  B  e.  H
)  /\  ( C  e.  H  /\  D  e.  H ) )  ->  E. z  z  =  S )
6 nfv 1466 . . 3  |-  F/ z ( ( A  e.  H  /\  B  e.  H )  /\  ( C  e.  H  /\  D  e.  H )
)
7 nfcv 2228 . . . . 5  |-  F/_ z <. A ,  B >.
8 ovi3.3 . . . . . 6  |-  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
) ) }
9 nfoprab3 5682 . . . . . 6  |-  F/_ z { <. <. 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
) ) }
108, 9nfcxfr 2225 . . . . 5  |-  F/_ z F
11 nfcv 2228 . . . . 5  |-  F/_ z <. C ,  D >.
127, 10, 11nfov 5661 . . . 4  |-  F/_ z
( <. A ,  B >. F <. C ,  D >. )
1312nfeq1 2238 . . 3  |-  F/ z ( <. A ,  B >. F <. C ,  D >. )  =  S
14 ovi3.2 . . . . . . 7  |-  ( ( ( w  =  A  /\  v  =  B )  /\  ( u  =  C  /\  f  =  D ) )  ->  R  =  S )
1514eqeq2d 2099 . . . . . 6  |-  ( ( ( w  =  A  /\  v  =  B )  /\  ( u  =  C  /\  f  =  D ) )  -> 
( z  =  R  <-> 
z  =  S ) )
1615copsex4g 4065 . . . . 5  |-  ( ( ( A  e.  H  /\  B  e.  H
)  /\  ( C  e.  H  /\  D  e.  H ) )  -> 
( E. w E. v E. u E. f
( ( <. A ,  B >.  =  <. w ,  v >.  /\  <. C ,  D >.  =  <. u ,  f >. )  /\  z  =  R
)  <->  z  =  S ) )
17 opelxpi 4459 . . . . . 6  |-  ( ( A  e.  H  /\  B  e.  H )  -> 
<. A ,  B >.  e.  ( H  X.  H
) )
18 opelxpi 4459 . . . . . 6  |-  ( ( C  e.  H  /\  D  e.  H )  -> 
<. C ,  D >.  e.  ( H  X.  H
) )
19 nfcv 2228 . . . . . . 7  |-  F/_ x <. A ,  B >.
20 nfcv 2228 . . . . . . 7  |-  F/_ y <. A ,  B >.
21 nfcv 2228 . . . . . . 7  |-  F/_ y <. C ,  D >.
22 nfv 1466 . . . . . . . 8  |-  F/ x E. w E. v E. u E. f ( ( <. A ,  B >.  =  <. w ,  v
>.  /\  y  =  <. u ,  f >. )  /\  z  =  R
)
23 nfoprab1 5680 . . . . . . . . . . 11  |-  F/_ x { <. <. 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
) ) }
248, 23nfcxfr 2225 . . . . . . . . . 10  |-  F/_ x F
25 nfcv 2228 . . . . . . . . . 10  |-  F/_ x
y
2619, 24, 25nfov 5661 . . . . . . . . 9  |-  F/_ x
( <. A ,  B >. F y )
2726nfeq1 2238 . . . . . . . 8  |-  F/ x
( <. A ,  B >. F y )  =  z
2822, 27nfim 1509 . . . . . . 7  |-  F/ x
( E. w E. v E. u E. f
( ( <. A ,  B >.  =  <. w ,  v >.  /\  y  =  <. u ,  f
>. )  /\  z  =  R )  ->  ( <. A ,  B >. F y )  =  z )
29 nfv 1466 . . . . . . . 8  |-  F/ y E. w E. v E. u E. f ( ( <. A ,  B >.  =  <. w ,  v
>.  /\  <. C ,  D >.  =  <. u ,  f
>. )  /\  z  =  R )
30 nfoprab2 5681 . . . . . . . . . . 11  |-  F/_ y { <. <. 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
) ) }
318, 30nfcxfr 2225 . . . . . . . . . 10  |-  F/_ y F
3220, 31, 21nfov 5661 . . . . . . . . 9  |-  F/_ y
( <. A ,  B >. F <. C ,  D >. )
3332nfeq1 2238 . . . . . . . 8  |-  F/ y ( <. A ,  B >. F <. C ,  D >. )  =  z
3429, 33nfim 1509 . . . . . . 7  |-  F/ y ( E. w E. v E. u E. f
( ( <. A ,  B >.  =  <. w ,  v >.  /\  <. C ,  D >.  =  <. u ,  f >. )  /\  z  =  R
)  ->  ( <. A ,  B >. F <. C ,  D >. )  =  z )
35 eqeq1 2094 . . . . . . . . . . 11  |-  ( x  =  <. A ,  B >.  ->  ( x  = 
<. w ,  v >.  <->  <. A ,  B >.  = 
<. w ,  v >.
) )
3635anbi1d 453 . . . . . . . . . 10  |-  ( x  =  <. A ,  B >.  ->  ( ( x  =  <. w ,  v
>.  /\  y  =  <. u ,  f >. )  <->  (
<. A ,  B >.  = 
<. w ,  v >.  /\  y  =  <. u ,  f >. )
) )
3736anbi1d 453 . . . . . . . . 9  |-  ( x  =  <. A ,  B >.  ->  ( ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f
>. )  /\  z  =  R )  <->  ( ( <. A ,  B >.  = 
<. w ,  v >.  /\  y  =  <. u ,  f >. )  /\  z  =  R
) ) )
38374exbidv 1798 . . . . . . . 8  |-  ( x  =  <. A ,  B >.  ->  ( E. w E. v E. u E. f ( ( x  =  <. w ,  v
>.  /\  y  =  <. u ,  f >. )  /\  z  =  R
)  <->  E. w E. v E. u E. f ( ( <. A ,  B >.  =  <. w ,  v
>.  /\  y  =  <. u ,  f >. )  /\  z  =  R
) ) )
39 oveq1 5641 . . . . . . . . 9  |-  ( x  =  <. A ,  B >.  ->  ( x F y )  =  (
<. A ,  B >. F y ) )
4039eqeq1d 2096 . . . . . . . 8  |-  ( x  =  <. A ,  B >.  ->  ( ( x F y )  =  z  <->  ( <. A ,  B >. F y )  =  z ) )
4138, 40imbi12d 232 . . . . . . 7  |-  ( x  =  <. A ,  B >.  ->  ( ( E. w E. v E. u E. f ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f
>. )  /\  z  =  R )  ->  (
x F y )  =  z )  <->  ( E. w E. v E. u E. f ( ( <. A ,  B >.  = 
<. w ,  v >.  /\  y  =  <. u ,  f >. )  /\  z  =  R
)  ->  ( <. A ,  B >. F y )  =  z ) ) )
42 eqeq1 2094 . . . . . . . . . . 11  |-  ( y  =  <. C ,  D >.  ->  ( y  = 
<. u ,  f >.  <->  <. C ,  D >.  = 
<. u ,  f >.
) )
4342anbi2d 452 . . . . . . . . . 10  |-  ( y  =  <. C ,  D >.  ->  ( ( <. A ,  B >.  = 
<. w ,  v >.  /\  y  =  <. u ,  f >. )  <->  (
<. A ,  B >.  = 
<. w ,  v >.  /\  <. C ,  D >.  =  <. u ,  f
>. ) ) )
4443anbi1d 453 . . . . . . . . 9  |-  ( y  =  <. C ,  D >.  ->  ( ( (
<. A ,  B >.  = 
<. w ,  v >.  /\  y  =  <. u ,  f >. )  /\  z  =  R
)  <->  ( ( <. A ,  B >.  = 
<. w ,  v >.  /\  <. C ,  D >.  =  <. u ,  f
>. )  /\  z  =  R ) ) )
45444exbidv 1798 . . . . . . . 8  |-  ( y  =  <. C ,  D >.  ->  ( E. w E. v E. u E. f ( ( <. A ,  B >.  = 
<. w ,  v >.  /\  y  =  <. u ,  f >. )  /\  z  =  R
)  <->  E. w E. v E. u E. f ( ( <. A ,  B >.  =  <. w ,  v
>.  /\  <. C ,  D >.  =  <. u ,  f
>. )  /\  z  =  R ) ) )
46 oveq2 5642 . . . . . . . . 9  |-  ( y  =  <. C ,  D >.  ->  ( <. A ,  B >. F y )  =  ( <. A ,  B >. F <. C ,  D >. ) )
4746eqeq1d 2096 . . . . . . . 8  |-  ( y  =  <. C ,  D >.  ->  ( ( <. A ,  B >. F y )  =  z  <-> 
( <. A ,  B >. F <. C ,  D >. )  =  z ) )
4845, 47imbi12d 232 . . . . . . 7  |-  ( y  =  <. C ,  D >.  ->  ( ( E. w E. v E. u E. f ( ( <. A ,  B >.  =  <. w ,  v
>.  /\  y  =  <. u ,  f >. )  /\  z  =  R
)  ->  ( <. A ,  B >. F y )  =  z )  <-> 
( E. w E. v E. u E. f
( ( <. A ,  B >.  =  <. w ,  v >.  /\  <. C ,  D >.  =  <. u ,  f >. )  /\  z  =  R
)  ->  ( <. A ,  B >. F <. C ,  D >. )  =  z ) ) )
49 moeq 2788 . . . . . . . . . . . 12  |-  E* z 
z  =  R
5049mosubop 4492 . . . . . . . . . . 11  |-  E* z E. u E. f ( y  =  <. u ,  f >.  /\  z  =  R )
5150mosubop 4492 . . . . . . . . . 10  |-  E* z E. w E. v ( x  =  <. w ,  v >.  /\  E. u E. f ( y  =  <. u ,  f
>.  /\  z  =  R ) )
52 anass 393 . . . . . . . . . . . . . 14  |-  ( ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f
>. )  /\  z  =  R )  <->  ( x  =  <. w ,  v
>.  /\  ( y  = 
<. u ,  f >.  /\  z  =  R
) ) )
53522exbii 1542 . . . . . . . . . . . . 13  |-  ( E. u E. f ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f
>. )  /\  z  =  R )  <->  E. u E. f ( x  = 
<. w ,  v >.  /\  ( y  =  <. u ,  f >.  /\  z  =  R ) ) )
54 19.42vv 1836 . . . . . . . . . . . . 13  |-  ( E. u E. f ( x  =  <. w ,  v >.  /\  (
y  =  <. u ,  f >.  /\  z  =  R ) )  <->  ( x  =  <. w ,  v
>.  /\  E. u E. f ( y  = 
<. u ,  f >.  /\  z  =  R
) ) )
5553, 54bitri 182 . . . . . . . . . . . 12  |-  ( E. u E. f ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f
>. )  /\  z  =  R )  <->  ( x  =  <. w ,  v
>.  /\  E. u E. f ( y  = 
<. u ,  f >.  /\  z  =  R
) ) )
56552exbii 1542 . . . . . . . . . . 11  |-  ( 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 ) ) )
5756mobii 1985 . . . . . . . . . 10  |-  ( 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 ) ) )
5851, 57mpbir 144 . . . . . . . . 9  |-  E* z E. w E. v E. u E. f ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f
>. )  /\  z  =  R )
5958a1i 9 . . . . . . . 8  |-  ( ( 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
) )
6059, 8ovidi 5745 . . . . . . 7  |-  ( ( 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
)  ->  ( x F y )  =  z ) )
6119, 20, 21, 28, 34, 41, 48, 60vtocl2gaf 2686 . . . . . 6  |-  ( (
<. A ,  B >.  e.  ( H  X.  H
)  /\  <. C ,  D >.  e.  ( H  X.  H ) )  ->  ( E. w E. v E. u E. f ( ( <. A ,  B >.  = 
<. w ,  v >.  /\  <. C ,  D >.  =  <. u ,  f
>. )  /\  z  =  R )  ->  ( <. A ,  B >. F
<. C ,  D >. )  =  z ) )
6217, 18, 61syl2an 283 . . . . 5  |-  ( ( ( A  e.  H  /\  B  e.  H
)  /\  ( C  e.  H  /\  D  e.  H ) )  -> 
( E. w E. v E. u E. f
( ( <. A ,  B >.  =  <. w ,  v >.  /\  <. C ,  D >.  =  <. u ,  f >. )  /\  z  =  R
)  ->  ( <. A ,  B >. F <. C ,  D >. )  =  z ) )
6316, 62sylbird 168 . . . 4  |-  ( ( ( A  e.  H  /\  B  e.  H
)  /\  ( C  e.  H  /\  D  e.  H ) )  -> 
( z  =  S  ->  ( <. A ,  B >. F <. C ,  D >. )  =  z ) )
64 eqeq2 2097 . . . 4  |-  ( z  =  S  ->  (
( <. A ,  B >. F <. C ,  D >. )  =  z  <->  ( <. A ,  B >. F <. C ,  D >. )  =  S ) )
6563, 64mpbidi 149 . . 3  |-  ( ( ( A  e.  H  /\  B  e.  H
)  /\  ( C  e.  H  /\  D  e.  H ) )  -> 
( z  =  S  ->  ( <. A ,  B >. F <. C ,  D >. )  =  S ) )
666, 13, 65exlimd 1533 . 2  |-  ( ( ( A  e.  H  /\  B  e.  H
)  /\  ( C  e.  H  /\  D  e.  H ) )  -> 
( E. z  z  =  S  ->  ( <. A ,  B >. F
<. C ,  D >. )  =  S ) )
675, 66mpd 13 1  |-  ( ( ( A  e.  H  /\  B  e.  H
)  /\  ( C  e.  H  /\  D  e.  H ) )  -> 
( <. A ,  B >. F <. C ,  D >. )  =  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289   E.wex 1426    e. wcel 1438   E*wmo 1949   _Vcvv 2619   <.cop 3444    X. cxp 4426  (class class class)co 5634   {coprab 5635
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-setind 4343
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-iota 4967  df-fun 5004  df-fv 5010  df-ov 5637  df-oprab 5638
This theorem is referenced by:  oviec  6378  addcnsr  7350  mulcnsr  7351
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