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Theorem ov3 6430
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 3050 . 2  |-  E. z 
z  =  S
3 nfv 1760 . . 3  |-  F/ z ( ( A  e.  H  /\  B  e.  H )  /\  ( C  e.  H  /\  D  e.  H )
)
4 nfcv 2591 . . . . 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 6339 . . . . . 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 2589 . . . . 5  |-  F/_ z F
8 nfcv 2591 . . . . 5  |-  F/_ z <. C ,  D >.
94, 7, 8nfov 6314 . . . 4  |-  F/_ z
( <. A ,  B >. F <. C ,  D >. )
109nfeq1 2604 . . 3  |-  F/ z ( <. A ,  B >. F <. C ,  D >. )  =  S
11 ov3.2 . . . . . . 7  |-  ( ( ( w  =  A  /\  v  =  B )  /\  ( u  =  C  /\  f  =  D ) )  ->  R  =  S )
1211eqeq2d 2460 . . . . . 6  |-  ( ( ( w  =  A  /\  v  =  B )  /\  ( u  =  C  /\  f  =  D ) )  -> 
( z  =  R  <-> 
z  =  S ) )
1312copsex4g 4689 . . . . 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 4865 . . . . . 6  |-  ( ( A  e.  H  /\  B  e.  H )  -> 
<. A ,  B >.  e.  ( H  X.  H
) )
15 opelxpi 4865 . . . . . 6  |-  ( ( C  e.  H  /\  D  e.  H )  -> 
<. C ,  D >.  e.  ( H  X.  H
) )
16 nfcv 2591 . . . . . . 7  |-  F/_ x <. A ,  B >.
17 nfcv 2591 . . . . . . 7  |-  F/_ y <. A ,  B >.
18 nfcv 2591 . . . . . . 7  |-  F/_ y <. C ,  D >.
19 nfv 1760 . . . . . . . 8  |-  F/ x E. w E. v E. u E. f ( ( <. A ,  B >.  =  <. w ,  v
>.  /\  y  =  <. u ,  f >. )  /\  z  =  R
)
20 nfoprab1 6337 . . . . . . . . . . 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 2589 . . . . . . . . . 10  |-  F/_ x F
22 nfcv 2591 . . . . . . . . . 10  |-  F/_ x
y
2316, 21, 22nfov 6314 . . . . . . . . 9  |-  F/_ x
( <. A ,  B >. F y )
2423nfeq1 2604 . . . . . . . 8  |-  F/ x
( <. A ,  B >. F y )  =  z
2519, 24nfim 2002 . . . . . . 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 1760 . . . . . . . 8  |-  F/ y E. w E. v E. u E. f ( ( <. A ,  B >.  =  <. w ,  v
>.  /\  <. C ,  D >.  =  <. u ,  f
>. )  /\  z  =  R )
27 nfoprab2 6338 . . . . . . . . . . 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 2589 . . . . . . . . . 10  |-  F/_ y F
2917, 28, 18nfov 6314 . . . . . . . . 9  |-  F/_ y
( <. A ,  B >. F <. C ,  D >. )
3029nfeq1 2604 . . . . . . . 8  |-  F/ y ( <. A ,  B >. F <. C ,  D >. )  =  z
3126, 30nfim 2002 . . . . . . 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 2454 . . . . . . . . . . 11  |-  ( x  =  <. A ,  B >.  ->  ( x  = 
<. w ,  v >.  <->  <. A ,  B >.  = 
<. w ,  v >.
) )
3332anbi1d 710 . . . . . . . . . 10  |-  ( x  =  <. A ,  B >.  ->  ( ( x  =  <. w ,  v
>.  /\  y  =  <. u ,  f >. )  <->  (
<. A ,  B >.  = 
<. w ,  v >.  /\  y  =  <. u ,  f >. )
) )
3433anbi1d 710 . . . . . . . . 9  |-  ( x  =  <. A ,  B >.  ->  ( ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f
>. )  /\  z  =  R )  <->  ( ( <. A ,  B >.  = 
<. w ,  v >.  /\  y  =  <. u ,  f >. )  /\  z  =  R
) ) )
35344exbidv 1771 . . . . . . . 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 6295 . . . . . . . . 9  |-  ( x  =  <. A ,  B >.  ->  ( x F y )  =  (
<. A ,  B >. F y ) )
3736eqeq1d 2452 . . . . . . . 8  |-  ( x  =  <. A ,  B >.  ->  ( ( x F y )  =  z  <->  ( <. A ,  B >. F y )  =  z ) )
3835, 37imbi12d 322 . . . . . . 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 2454 . . . . . . . . . . 11  |-  ( y  =  <. C ,  D >.  ->  ( y  = 
<. u ,  f >.  <->  <. C ,  D >.  = 
<. u ,  f >.
) )
4039anbi2d 709 . . . . . . . . . 10  |-  ( y  =  <. C ,  D >.  ->  ( ( <. A ,  B >.  = 
<. w ,  v >.  /\  y  =  <. u ,  f >. )  <->  (
<. A ,  B >.  = 
<. w ,  v >.  /\  <. C ,  D >.  =  <. u ,  f
>. ) ) )
4140anbi1d 710 . . . . . . . . 9  |-  ( y  =  <. C ,  D >.  ->  ( ( (
<. A ,  B >.  = 
<. w ,  v >.  /\  y  =  <. u ,  f >. )  /\  z  =  R
)  <->  ( ( <. A ,  B >.  = 
<. w ,  v >.  /\  <. C ,  D >.  =  <. u ,  f
>. )  /\  z  =  R ) ) )
42414exbidv 1771 . . . . . . . 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 6296 . . . . . . . . 9  |-  ( y  =  <. C ,  D >.  ->  ( <. A ,  B >. F y )  =  ( <. A ,  B >. F <. C ,  D >. ) )
4443eqeq1d 2452 . . . . . . . 8  |-  ( y  =  <. C ,  D >.  ->  ( ( <. A ,  B >. F y )  =  z  <-> 
( <. A ,  B >. F <. C ,  D >. )  =  z ) )
4542, 44imbi12d 322 . . . . . . 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 3213 . . . . . . . . . . . 12  |-  E* z 
z  =  R
4746mosubop 4699 . . . . . . . . . . 11  |-  E* z E. u E. f ( y  =  <. u ,  f >.  /\  z  =  R )
4847mosubop 4699 . . . . . . . . . 10  |-  E* z E. w E. v ( x  =  <. w ,  v >.  /\  E. u E. f ( y  =  <. u ,  f
>.  /\  z  =  R ) )
49 anass 654 . . . . . . . . . . . . . 14  |-  ( ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f
>. )  /\  z  =  R )  <->  ( x  =  <. w ,  v
>.  /\  ( y  = 
<. u ,  f >.  /\  z  =  R
) ) )
50492exbii 1718 . . . . . . . . . . . . 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 1835 . . . . . . . . . . . . 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 253 . . . . . . . . . . . 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 1718 . . . . . . . . . . 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 2321 . . . . . . . . . 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 213 . . . . . . . . 9  |-  E* z E. w E. v E. u E. f ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f
>. )  /\  z  =  R )
5655a1i 11 . . . . . . . 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 6412 . . . . . . 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 3113 . . . . . 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 480 . . . . 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 239 . . . 4  |-  ( ( ( A  e.  H  /\  B  e.  H
)  /\  ( C  e.  H  /\  D  e.  H ) )  -> 
( z  =  S  ->  ( <. A ,  B >. F <. C ,  D >. )  =  z ) )
61 eqeq2 2461 . . . 4  |-  ( z  =  S  ->  (
( <. A ,  B >. F <. C ,  D >. )  =  z  <->  ( <. A ,  B >. F <. C ,  D >. )  =  S ) )
6260, 61mpbidi 220 . . 3  |-  ( ( ( A  e.  H  /\  B  e.  H
)  /\  ( C  e.  H  /\  D  e.  H ) )  -> 
( z  =  S  ->  ( <. A ,  B >. F <. C ,  D >. )  =  S ) )
633, 10, 62exlimd 1996 . 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 20 1  |-  ( ( ( A  e.  H  /\  B  e.  H
)  /\  ( C  e.  H  /\  D  e.  H ) )  -> 
( <. A ,  B >. F <. C ,  D >. )  =  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1443   E.wex 1662    e. wcel 1886   E*wmo 2299   _Vcvv 3044   <.cop 3973    X. cxp 4831  (class class class)co 6288   {coprab 6289
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pr 4638
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-sbc 3267  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-br 4402  df-opab 4461  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-iota 5545  df-fun 5583  df-fv 5589  df-ov 6291  df-oprab 6292
This theorem is referenced by:  addcnsr  9556  mulcnsr  9557
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