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Theorem ov6g 6012
Description: The value of an operation class abstraction. Special case. (Contributed by NM, 13-Nov-2006.)
Hypotheses
Ref Expression
ov6g.1  |-  ( <.
x ,  y >.  =  <. A ,  B >.  ->  R  =  S )
ov6g.2  |-  F  =  { <. <. x ,  y
>. ,  z >.  |  ( <. x ,  y
>.  e.  C  /\  z  =  R ) }
Assertion
Ref Expression
ov6g  |-  ( ( ( A  e.  G  /\  B  e.  H  /\  <. A ,  B >.  e.  C )  /\  S  e.  J )  ->  ( A F B )  =  S )
Distinct variable groups:    x, y, z, A    x, B, y, z    x, C, y, z    z, R    x, S, y, z
Allowed substitution hints:    R( x, y)    F( x, y, z)    G( x, y, z)    H( x, y, z)    J( x, y, z)

Proof of Theorem ov6g
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 df-ov 5878 . 2  |-  ( A F B )  =  ( F `  <. A ,  B >. )
2 eqid 2177 . . . . . 6  |-  S  =  S
3 biidd 172 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  ( S  =  S  <-> 
S  =  S ) )
43copsex2g 4247 . . . . . 6  |-  ( ( A  e.  G  /\  B  e.  H )  ->  ( E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  S  =  S )  <->  S  =  S ) )
52, 4mpbiri 168 . . . . 5  |-  ( ( A  e.  G  /\  B  e.  H )  ->  E. x E. y
( <. A ,  B >.  =  <. x ,  y
>.  /\  S  =  S ) )
653adant3 1017 . . . 4  |-  ( ( A  e.  G  /\  B  e.  H  /\  <. A ,  B >.  e.  C )  ->  E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  S  =  S ) )
76adantr 276 . . 3  |-  ( ( ( A  e.  G  /\  B  e.  H  /\  <. A ,  B >.  e.  C )  /\  S  e.  J )  ->  E. x E. y
( <. A ,  B >.  =  <. x ,  y
>.  /\  S  =  S ) )
8 eqeq1 2184 . . . . . . . 8  |-  ( w  =  <. A ,  B >.  ->  ( w  = 
<. x ,  y >.  <->  <. A ,  B >.  = 
<. x ,  y >.
) )
98anbi1d 465 . . . . . . 7  |-  ( w  =  <. A ,  B >.  ->  ( ( w  =  <. x ,  y
>.  /\  z  =  R )  <->  ( <. A ,  B >.  =  <. x ,  y >.  /\  z  =  R ) ) )
10 ov6g.1 . . . . . . . . . 10  |-  ( <.
x ,  y >.  =  <. A ,  B >.  ->  R  =  S )
1110eqeq2d 2189 . . . . . . . . 9  |-  ( <.
x ,  y >.  =  <. A ,  B >.  ->  ( z  =  R  <->  z  =  S ) )
1211eqcoms 2180 . . . . . . . 8  |-  ( <. A ,  B >.  = 
<. x ,  y >.  ->  ( z  =  R  <-> 
z  =  S ) )
1312pm5.32i 454 . . . . . . 7  |-  ( (
<. A ,  B >.  = 
<. x ,  y >.  /\  z  =  R
)  <->  ( <. A ,  B >.  =  <. x ,  y >.  /\  z  =  S ) )
149, 13bitrdi 196 . . . . . 6  |-  ( w  =  <. A ,  B >.  ->  ( ( w  =  <. x ,  y
>.  /\  z  =  R )  <->  ( <. A ,  B >.  =  <. x ,  y >.  /\  z  =  S ) ) )
15142exbidv 1868 . . . . 5  |-  ( w  =  <. A ,  B >.  ->  ( E. x E. y ( w  = 
<. x ,  y >.  /\  z  =  R
)  <->  E. x E. y
( <. A ,  B >.  =  <. x ,  y
>.  /\  z  =  S ) ) )
16 eqeq1 2184 . . . . . . 7  |-  ( z  =  S  ->  (
z  =  S  <->  S  =  S ) )
1716anbi2d 464 . . . . . 6  |-  ( z  =  S  ->  (
( <. A ,  B >.  =  <. x ,  y
>.  /\  z  =  S )  <->  ( <. A ,  B >.  =  <. x ,  y >.  /\  S  =  S ) ) )
18172exbidv 1868 . . . . 5  |-  ( z  =  S  ->  ( E. x E. y (
<. A ,  B >.  = 
<. x ,  y >.  /\  z  =  S
)  <->  E. x E. y
( <. A ,  B >.  =  <. x ,  y
>.  /\  S  =  S ) ) )
19 moeq 2913 . . . . . . 7  |-  E* z 
z  =  R
2019mosubop 4693 . . . . . 6  |-  E* z E. x E. y ( w  =  <. x ,  y >.  /\  z  =  R )
2120a1i 9 . . . . 5  |-  ( w  e.  C  ->  E* z E. x E. y
( w  =  <. x ,  y >.  /\  z  =  R ) )
22 ov6g.2 . . . . . 6  |-  F  =  { <. <. x ,  y
>. ,  z >.  |  ( <. x ,  y
>.  e.  C  /\  z  =  R ) }
23 dfoprab2 5922 . . . . . 6  |-  { <. <.
x ,  y >. ,  z >.  |  (
<. x ,  y >.  e.  C  /\  z  =  R ) }  =  { <. w ,  z
>.  |  E. x E. y ( w  = 
<. x ,  y >.  /\  ( <. x ,  y
>.  e.  C  /\  z  =  R ) ) }
24 eleq1 2240 . . . . . . . . . . . 12  |-  ( w  =  <. x ,  y
>.  ->  ( w  e.  C  <->  <. x ,  y
>.  e.  C ) )
2524anbi1d 465 . . . . . . . . . . 11  |-  ( w  =  <. x ,  y
>.  ->  ( ( w  e.  C  /\  z  =  R )  <->  ( <. x ,  y >.  e.  C  /\  z  =  R
) ) )
2625pm5.32i 454 . . . . . . . . . 10  |-  ( ( w  =  <. x ,  y >.  /\  (
w  e.  C  /\  z  =  R )
)  <->  ( w  = 
<. x ,  y >.  /\  ( <. x ,  y
>.  e.  C  /\  z  =  R ) ) )
27 an12 561 . . . . . . . . . 10  |-  ( ( w  =  <. x ,  y >.  /\  (
w  e.  C  /\  z  =  R )
)  <->  ( w  e.  C  /\  ( w  =  <. x ,  y
>.  /\  z  =  R ) ) )
2826, 27bitr3i 186 . . . . . . . . 9  |-  ( ( w  =  <. x ,  y >.  /\  ( <. x ,  y >.  e.  C  /\  z  =  R ) )  <->  ( w  e.  C  /\  (
w  =  <. x ,  y >.  /\  z  =  R ) ) )
29282exbii 1606 . . . . . . . 8  |-  ( E. x E. y ( w  =  <. x ,  y >.  /\  ( <. x ,  y >.  e.  C  /\  z  =  R ) )  <->  E. x E. y ( w  e.  C  /\  ( w  =  <. x ,  y
>.  /\  z  =  R ) ) )
30 19.42vv 1911 . . . . . . . 8  |-  ( E. x E. y ( w  e.  C  /\  ( w  =  <. x ,  y >.  /\  z  =  R ) )  <->  ( w  e.  C  /\  E. x E. y ( w  = 
<. x ,  y >.  /\  z  =  R
) ) )
3129, 30bitri 184 . . . . . . 7  |-  ( E. x E. y ( w  =  <. x ,  y >.  /\  ( <. x ,  y >.  e.  C  /\  z  =  R ) )  <->  ( w  e.  C  /\  E. x E. y ( w  = 
<. x ,  y >.  /\  z  =  R
) ) )
3231opabbii 4071 . . . . . 6  |-  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ( <. x ,  y >.  e.  C  /\  z  =  R
) ) }  =  { <. w ,  z
>.  |  ( w  e.  C  /\  E. x E. y ( w  = 
<. x ,  y >.  /\  z  =  R
) ) }
3322, 23, 323eqtri 2202 . . . . 5  |-  F  =  { <. w ,  z
>.  |  ( w  e.  C  /\  E. x E. y ( w  = 
<. x ,  y >.  /\  z  =  R
) ) }
3415, 18, 21, 33fvopab3ig 5591 . . . 4  |-  ( (
<. A ,  B >.  e.  C  /\  S  e.  J )  ->  ( E. x E. y (
<. A ,  B >.  = 
<. x ,  y >.  /\  S  =  S
)  ->  ( F `  <. A ,  B >. )  =  S ) )
35343ad2antl3 1161 . . 3  |-  ( ( ( A  e.  G  /\  B  e.  H  /\  <. A ,  B >.  e.  C )  /\  S  e.  J )  ->  ( E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  S  =  S )  ->  ( F `  <. A ,  B >. )  =  S ) )
367, 35mpd 13 . 2  |-  ( ( ( A  e.  G  /\  B  e.  H  /\  <. A ,  B >.  e.  C )  /\  S  e.  J )  ->  ( F `  <. A ,  B >. )  =  S )
371, 36eqtrid 2222 1  |-  ( ( ( A  e.  G  /\  B  e.  H  /\  <. A ,  B >.  e.  C )  /\  S  e.  J )  ->  ( A F B )  =  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 978    = wceq 1353   E.wex 1492   E*wmo 2027    e. wcel 2148   <.cop 3596   {copab 4064   ` cfv 5217  (class class class)co 5875   {coprab 5876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-sbc 2964  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-iota 5179  df-fun 5219  df-fv 5225  df-ov 5878  df-oprab 5879
This theorem is referenced by: (None)
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