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Theorem ov3 6110
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
ov3.1  |-  S  e. 
_V
ov3.2  |-  ( ( ( w  =  A  /\  v  =  B )  /\  ( u  =  C  /\  f  =  D ) )  ->  R  =  S )
ov3.3  |-  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
ov3  |-  ( ( ( 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 ov3
StepHypRef Expression
1 ov3.1 . . 3  |-  S  e. 
_V
21isseti 2879 . 2  |-  E. z 
z  =  S
3 nfv 1624 . . 3  |-  F/ z ( ( A  e.  H  /\  B  e.  H )  /\  ( C  e.  H  /\  D  e.  H )
)
4 nfcv 2502 . . . . 5  |-  F/_ z <. A ,  B >.
5 ov3.3 . . . . . 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
) ) }
6 nfoprab3 6025 . . . . . 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
) ) }
75, 6nfcxfr 2499 . . . . 5  |-  F/_ z F
8 nfcv 2502 . . . . 5  |-  F/_ z <. C ,  D >.
94, 7, 8nfov 6004 . . . 4  |-  F/_ z
( <. A ,  B >. F <. C ,  D >. )
109nfeq1 2511 . . 3  |-  F/ z ( <. A ,  B >. F <. C ,  D >. )  =  S
11 ov3.2 . . . . . . 7  |-  ( ( ( w  =  A  /\  v  =  B )  /\  ( u  =  C  /\  f  =  D ) )  ->  R  =  S )
1211eqeq2d 2377 . . . . . 6  |-  ( ( ( w  =  A  /\  v  =  B )  /\  ( u  =  C  /\  f  =  D ) )  -> 
( z  =  R  <-> 
z  =  S ) )
1312copsex4g 4358 . . . . 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 ) )
14 opelxpi 4824 . . . . . 6  |-  ( ( A  e.  H  /\  B  e.  H )  -> 
<. A ,  B >.  e.  ( H  X.  H
) )
15 opelxpi 4824 . . . . . 6  |-  ( ( C  e.  H  /\  D  e.  H )  -> 
<. C ,  D >.  e.  ( H  X.  H
) )
16 nfcv 2502 . . . . . . 7  |-  F/_ x <. A ,  B >.
17 nfcv 2502 . . . . . . 7  |-  F/_ y <. A ,  B >.
18 nfcv 2502 . . . . . . 7  |-  F/_ y <. C ,  D >.
19 nfv 1624 . . . . . . . 8  |-  F/ x E. w E. v E. u E. f ( ( <. A ,  B >.  =  <. w ,  v
>.  /\  y  =  <. u ,  f >. )  /\  z  =  R
)
20 nfoprab1 6023 . . . . . . . . . . 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
) ) }
215, 20nfcxfr 2499 . . . . . . . . . 10  |-  F/_ x F
22 nfcv 2502 . . . . . . . . . 10  |-  F/_ x
y
2316, 21, 22nfov 6004 . . . . . . . . 9  |-  F/_ x
( <. A ,  B >. F y )
2423nfeq1 2511 . . . . . . . 8  |-  F/ x
( <. A ,  B >. F y )  =  z
2519, 24nfim 1820 . . . . . . 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 )
26 nfv 1624 . . . . . . . 8  |-  F/ y E. w E. v E. u E. f ( ( <. A ,  B >.  =  <. w ,  v
>.  /\  <. C ,  D >.  =  <. u ,  f
>. )  /\  z  =  R )
27 nfoprab2 6024 . . . . . . . . . . 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
) ) }
285, 27nfcxfr 2499 . . . . . . . . . 10  |-  F/_ y F
2917, 28, 18nfov 6004 . . . . . . . . 9  |-  F/_ y
( <. A ,  B >. F <. C ,  D >. )
3029nfeq1 2511 . . . . . . . 8  |-  F/ y ( <. A ,  B >. F <. C ,  D >. )  =  z
3126, 30nfim 1820 . . . . . . 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 )
32 eqeq1 2372 . . . . . . . . . . 11  |-  ( x  =  <. A ,  B >.  ->  ( x  = 
<. w ,  v >.  <->  <. A ,  B >.  = 
<. w ,  v >.
) )
3332anbi1d 685 . . . . . . . . . 10  |-  ( x  =  <. A ,  B >.  ->  ( ( x  =  <. w ,  v
>.  /\  y  =  <. u ,  f >. )  <->  (
<. A ,  B >.  = 
<. w ,  v >.  /\  y  =  <. u ,  f >. )
) )
3433anbi1d 685 . . . . . . . . 9  |-  ( x  =  <. A ,  B >.  ->  ( ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f
>. )  /\  z  =  R )  <->  ( ( <. A ,  B >.  = 
<. w ,  v >.  /\  y  =  <. u ,  f >. )  /\  z  =  R
) ) )
35344exbidv 1635 . . . . . . . 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
) ) )
36 oveq1 5988 . . . . . . . . 9  |-  ( x  =  <. A ,  B >.  ->  ( x F y )  =  (
<. A ,  B >. F y ) )
3736eqeq1d 2374 . . . . . . . 8  |-  ( x  =  <. A ,  B >.  ->  ( ( x F y )  =  z  <->  ( <. A ,  B >. F y )  =  z ) )
3835, 37imbi12d 311 . . . . . . 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 ) ) )
39 eqeq1 2372 . . . . . . . . . . 11  |-  ( y  =  <. C ,  D >.  ->  ( y  = 
<. u ,  f >.  <->  <. C ,  D >.  = 
<. u ,  f >.
) )
4039anbi2d 684 . . . . . . . . . 10  |-  ( y  =  <. C ,  D >.  ->  ( ( <. A ,  B >.  = 
<. w ,  v >.  /\  y  =  <. u ,  f >. )  <->  (
<. A ,  B >.  = 
<. w ,  v >.  /\  <. C ,  D >.  =  <. u ,  f
>. ) ) )
4140anbi1d 685 . . . . . . . . 9  |-  ( y  =  <. C ,  D >.  ->  ( ( (
<. A ,  B >.  = 
<. w ,  v >.  /\  y  =  <. u ,  f >. )  /\  z  =  R
)  <->  ( ( <. A ,  B >.  = 
<. w ,  v >.  /\  <. C ,  D >.  =  <. u ,  f
>. )  /\  z  =  R ) ) )
42414exbidv 1635 . . . . . . . 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 ) ) )
43 oveq2 5989 . . . . . . . . 9  |-  ( y  =  <. C ,  D >.  ->  ( <. A ,  B >. F y )  =  ( <. A ,  B >. F <. C ,  D >. ) )
4443eqeq1d 2374 . . . . . . . 8  |-  ( y  =  <. C ,  D >.  ->  ( ( <. A ,  B >. F y )  =  z  <-> 
( <. A ,  B >. F <. C ,  D >. )  =  z ) )
4542, 44imbi12d 311 . . . . . . 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 ) ) )
46 moeq 3027 . . . . . . . . . . . 12  |-  E* z 
z  =  R
4746mosubop 4368 . . . . . . . . . . 11  |-  E* z E. u E. f ( y  =  <. u ,  f >.  /\  z  =  R )
4847mosubop 4368 . . . . . . . . . 10  |-  E* z E. w E. v ( x  =  <. w ,  v >.  /\  E. u E. f ( y  =  <. u ,  f
>.  /\  z  =  R ) )
49 anass 630 . . . . . . . . . . . . . 14  |-  ( ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f
>. )  /\  z  =  R )  <->  ( x  =  <. w ,  v
>.  /\  ( y  = 
<. u ,  f >.  /\  z  =  R
) ) )
50492exbii 1588 . . . . . . . . . . . . 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 ) ) )
51 19.42vv 1917 . . . . . . . . . . . . 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
) ) )
5250, 51bitri 240 . . . . . . . . . . . 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
) ) )
53522exbii 1588 . . . . . . . . . . 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 ) ) )
5453mobii 2253 . . . . . . . . . 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 ) ) )
5548, 54mpbir 200 . . . . . . . . 9  |-  E* z E. w E. v E. u E. f ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f
>. )  /\  z  =  R )
5655a1i 10 . . . . . . . 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
) )
5756, 5ovidi 6092 . . . . . . 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 ) )
5816, 17, 18, 25, 31, 38, 45, 57vtocl2gaf 2935 . . . . . 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 ) )
5914, 15, 58syl2an 463 . . . . 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 ) )
6013, 59sylbird 226 . . . 4  |-  ( ( ( A  e.  H  /\  B  e.  H
)  /\  ( C  e.  H  /\  D  e.  H ) )  -> 
( z  =  S  ->  ( <. A ,  B >. F <. C ,  D >. )  =  z ) )
61 eqeq2 2375 . . . 4  |-  ( z  =  S  ->  (
( <. A ,  B >. F <. C ,  D >. )  =  z  <->  ( <. A ,  B >. F <. C ,  D >. )  =  S ) )
6260, 61mpbidi 207 . . 3  |-  ( ( ( A  e.  H  /\  B  e.  H
)  /\  ( C  e.  H  /\  D  e.  H ) )  -> 
( z  =  S  ->  ( <. A ,  B >. F <. C ,  D >. )  =  S ) )
633, 10, 62exlimd 1812 . 2  |-  ( ( ( A  e.  H  /\  B  e.  H
)  /\  ( C  e.  H  /\  D  e.  H ) )  -> 
( E. z  z  =  S  ->  ( <. A ,  B >. F
<. C ,  D >. )  =  S ) )
642, 63mpi 16 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 358   E.wex 1546    = wceq 1647    e. wcel 1715   E*wmo 2218   _Vcvv 2873   <.cop 3732    X. cxp 4790  (class class class)co 5981   {coprab 5982
This theorem is referenced by:  ovec  6911  addcnsr  8904  mulcnsr  8905
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-iota 5322  df-fun 5360  df-fv 5366  df-ov 5984  df-oprab 5985
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