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Theorem ov6g 6001
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 5877 . 2  |-  ( A F B )  =  ( F `  <. A ,  B >. )
2 eqid 2296 . . . . . 6  |-  S  =  S
3 biidd 228 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  ( S  =  S  <-> 
S  =  S ) )
43copsex2g 4270 . . . . . 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 2302 . . . . . . . 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 2307 . . . . . . . . 9  |-  ( <.
x ,  y >.  =  <. A ,  B >.  ->  ( z  =  R  <->  z  =  S ) )
1211eqcoms 2299 . . . . . . . 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 1618 . . . . 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 2302 . . . . . . 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 1618 . . . . 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 2954 . . . . . . 7  |-  E* z 
z  =  R
2019mosubop 4281 . . . . . 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 5911 . . . . . 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 2356 . . . . . . . . . . . 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 1573 . . . . . . . 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 1860 . . . . . . . 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 4099 . . . . . 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 2320 . . . . 5  |-  F  =  { <. w ,  z
>.  |  ( w  e.  C  /\  E. x E. y ( w  = 
<. x ,  y >.  /\  z  =  R
) ) }
3415, 18, 21, 33fvopab3ig 5615 . . . 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 2340 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 1531    = wceq 1632    e. wcel 1696   E*wmo 2157   <.cop 3656   {copab 4092   ` cfv 5271  (class class class)co 5874   {coprab 5875
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878
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