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Theorem ov6g 5985
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 5861 . 2  |-  ( A F B )  =  ( F `  <. A ,  B >. )
2 eqid 2283 . . . . . 6  |-  S  =  S
3 biidd 228 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  ( S  =  S  <-> 
S  =  S ) )
43copsex2g 4254 . . . . . 6  |-  ( ( A  e.  G  /\  B  e.  H )  ->  ( E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  S  =  S )  <->  S  =  S ) )
52, 4mpbiri 224 . . . . 5  |-  ( ( A  e.  G  /\  B  e.  H )  ->  E. x E. y
( <. A ,  B >.  =  <. x ,  y
>.  /\  S  =  S ) )
653adant3 975 . . . 4  |-  ( ( A  e.  G  /\  B  e.  H  /\  <. A ,  B >.  e.  C )  ->  E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  S  =  S ) )
76adantr 451 . . 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 2289 . . . . . . . 8  |-  ( w  =  <. A ,  B >.  ->  ( w  = 
<. x ,  y >.  <->  <. A ,  B >.  = 
<. x ,  y >.
) )
98anbi1d 685 . . . . . . 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 2294 . . . . . . . . 9  |-  ( <.
x ,  y >.  =  <. A ,  B >.  ->  ( z  =  R  <->  z  =  S ) )
1211eqcoms 2286 . . . . . . . 8  |-  ( <. A ,  B >.  = 
<. x ,  y >.  ->  ( z  =  R  <-> 
z  =  S ) )
1312pm5.32i 618 . . . . . . 7  |-  ( (
<. A ,  B >.  = 
<. x ,  y >.  /\  z  =  R
)  <->  ( <. A ,  B >.  =  <. x ,  y >.  /\  z  =  S ) )
149, 13syl6bb 252 . . . . . 6  |-  ( w  =  <. A ,  B >.  ->  ( ( w  =  <. x ,  y
>.  /\  z  =  R )  <->  ( <. A ,  B >.  =  <. x ,  y >.  /\  z  =  S ) ) )
15142exbidv 1614 . . . . 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 2289 . . . . . . 7  |-  ( z  =  S  ->  (
z  =  S  <->  S  =  S ) )
1716anbi2d 684 . . . . . 6  |-  ( z  =  S  ->  (
( <. A ,  B >.  =  <. x ,  y
>.  /\  z  =  S )  <->  ( <. A ,  B >.  =  <. x ,  y >.  /\  S  =  S ) ) )
18172exbidv 1614 . . . . 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 2941 . . . . . . 7  |-  E* z 
z  =  R
2019mosubop 4265 . . . . . 6  |-  E* z E. x E. y ( w  =  <. x ,  y >.  /\  z  =  R )
2120a1i 10 . . . . 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 5895 . . . . . 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 2343 . . . . . . . . . . . 12  |-  ( w  =  <. x ,  y
>.  ->  ( w  e.  C  <->  <. x ,  y
>.  e.  C ) )
2524anbi1d 685 . . . . . . . . . . 11  |-  ( w  =  <. x ,  y
>.  ->  ( ( w  e.  C  /\  z  =  R )  <->  ( <. x ,  y >.  e.  C  /\  z  =  R
) ) )
2625pm5.32i 618 . . . . . . . . . 10  |-  ( ( w  =  <. x ,  y >.  /\  (
w  e.  C  /\  z  =  R )
)  <->  ( w  = 
<. x ,  y >.  /\  ( <. x ,  y
>.  e.  C  /\  z  =  R ) ) )
27 an12 772 . . . . . . . . . 10  |-  ( ( w  =  <. x ,  y >.  /\  (
w  e.  C  /\  z  =  R )
)  <->  ( w  e.  C  /\  ( w  =  <. x ,  y
>.  /\  z  =  R ) ) )
2826, 27bitr3i 242 . . . . . . . . 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 1848 . . . . . . . 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 240 . . . . . . 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 4083 . . . . . 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 2307 . . . . 5  |-  F  =  { <. w ,  z
>.  |  ( w  e.  C  /\  E. x E. y ( w  = 
<. x ,  y >.  /\  z  =  R
) ) }
3415, 18, 21, 33fvopab3ig 5599 . . . 4  |-  ( (
<. A ,  B >.  e.  C  /\  S  e.  J )  ->  ( E. x E. y (
<. A ,  B >.  = 
<. x ,  y >.  /\  S  =  S
)  ->  ( F `  <. A ,  B >. )  =  S ) )
35343ad2antl3 1119 . . 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 14 . 2  |-  ( ( ( A  e.  G  /\  B  e.  H  /\  <. A ,  B >.  e.  C )  /\  S  e.  J )  ->  ( F `  <. A ,  B >. )  =  S )
371, 36syl5eq 2327 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    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684   E*wmo 2144   <.cop 3643   {copab 4076   ` cfv 5255  (class class class)co 5858   {coprab 5859
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862
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