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Theorem ov6g 5876
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 5745 . 2  |-  ( A F B )  =  ( F `  <. A ,  B >. )
2 eqid 2117 . . . . . 6  |-  S  =  S
3 biidd 171 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  ( S  =  S  <-> 
S  =  S ) )
43copsex2g 4138 . . . . . 6  |-  ( ( A  e.  G  /\  B  e.  H )  ->  ( E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  S  =  S )  <->  S  =  S ) )
52, 4mpbiri 167 . . . . 5  |-  ( ( A  e.  G  /\  B  e.  H )  ->  E. x E. y
( <. A ,  B >.  =  <. x ,  y
>.  /\  S  =  S ) )
653adant3 986 . . . 4  |-  ( ( A  e.  G  /\  B  e.  H  /\  <. A ,  B >.  e.  C )  ->  E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  S  =  S ) )
76adantr 274 . . 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 2124 . . . . . . . 8  |-  ( w  =  <. A ,  B >.  ->  ( w  = 
<. x ,  y >.  <->  <. A ,  B >.  = 
<. x ,  y >.
) )
98anbi1d 460 . . . . . . 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 2129 . . . . . . . . 9  |-  ( <.
x ,  y >.  =  <. A ,  B >.  ->  ( z  =  R  <->  z  =  S ) )
1211eqcoms 2120 . . . . . . . 8  |-  ( <. A ,  B >.  = 
<. x ,  y >.  ->  ( z  =  R  <-> 
z  =  S ) )
1312pm5.32i 449 . . . . . . 7  |-  ( (
<. A ,  B >.  = 
<. x ,  y >.  /\  z  =  R
)  <->  ( <. A ,  B >.  =  <. x ,  y >.  /\  z  =  S ) )
149, 13syl6bb 195 . . . . . 6  |-  ( w  =  <. A ,  B >.  ->  ( ( w  =  <. x ,  y
>.  /\  z  =  R )  <->  ( <. A ,  B >.  =  <. x ,  y >.  /\  z  =  S ) ) )
15142exbidv 1824 . . . . 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 2124 . . . . . . 7  |-  ( z  =  S  ->  (
z  =  S  <->  S  =  S ) )
1716anbi2d 459 . . . . . 6  |-  ( z  =  S  ->  (
( <. A ,  B >.  =  <. x ,  y
>.  /\  z  =  S )  <->  ( <. A ,  B >.  =  <. x ,  y >.  /\  S  =  S ) ) )
18172exbidv 1824 . . . . 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 2832 . . . . . . 7  |-  E* z 
z  =  R
2019mosubop 4575 . . . . . 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 5786 . . . . . 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 2180 . . . . . . . . . . . 12  |-  ( w  =  <. x ,  y
>.  ->  ( w  e.  C  <->  <. x ,  y
>.  e.  C ) )
2524anbi1d 460 . . . . . . . . . . 11  |-  ( w  =  <. x ,  y
>.  ->  ( ( w  e.  C  /\  z  =  R )  <->  ( <. x ,  y >.  e.  C  /\  z  =  R
) ) )
2625pm5.32i 449 . . . . . . . . . 10  |-  ( ( w  =  <. x ,  y >.  /\  (
w  e.  C  /\  z  =  R )
)  <->  ( w  = 
<. x ,  y >.  /\  ( <. x ,  y
>.  e.  C  /\  z  =  R ) ) )
27 an12 535 . . . . . . . . . 10  |-  ( ( w  =  <. x ,  y >.  /\  (
w  e.  C  /\  z  =  R )
)  <->  ( w  e.  C  /\  ( w  =  <. x ,  y
>.  /\  z  =  R ) ) )
2826, 27bitr3i 185 . . . . . . . . 9  |-  ( ( w  =  <. x ,  y >.  /\  ( <. x ,  y >.  e.  C  /\  z  =  R ) )  <->  ( w  e.  C  /\  (
w  =  <. x ,  y >.  /\  z  =  R ) ) )
29282exbii 1570 . . . . . . . 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 1865 . . . . . . . 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 183 . . . . . . 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 3965 . . . . . 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 2142 . . . . 5  |-  F  =  { <. w ,  z
>.  |  ( w  e.  C  /\  E. x E. y ( w  = 
<. x ,  y >.  /\  z  =  R
) ) }
3415, 18, 21, 33fvopab3ig 5463 . . . 4  |-  ( (
<. A ,  B >.  e.  C  /\  S  e.  J )  ->  ( E. x E. y (
<. A ,  B >.  = 
<. x ,  y >.  /\  S  =  S
)  ->  ( F `  <. A ,  B >. )  =  S ) )
35343ad2antl3 1130 . . 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, 36syl5eq 2162 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 103    <-> wb 104    /\ w3a 947    = wceq 1316   E.wex 1453    e. wcel 1465   E*wmo 1978   <.cop 3500   {copab 3958   ` cfv 5093  (class class class)co 5742   {coprab 5743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fv 5101  df-ov 5745  df-oprab 5746
This theorem is referenced by: (None)
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