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Theorem offval3 5813
Description: General value of  ( F  oF R G ) with no assumptions on functionality of  F and  G. (Contributed by Stefan O'Rear, 24-Jan-2015.)
Assertion
Ref Expression
offval3  |-  ( ( F  e.  V  /\  G  e.  W )  ->  ( F  oF R G )  =  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( ( F `
 x ) R ( G `  x
) ) ) )
Distinct variable groups:    x, F    x, G    x, V    x, W    x, R

Proof of Theorem offval3
Dummy variables  a  b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2619 . . 3  |-  ( F  e.  V  ->  F  e.  _V )
21adantr 270 . 2  |-  ( ( F  e.  V  /\  G  e.  W )  ->  F  e.  _V )
3 elex 2619 . . 3  |-  ( G  e.  W  ->  G  e.  _V )
43adantl 271 . 2  |-  ( ( F  e.  V  /\  G  e.  W )  ->  G  e.  _V )
5 dmexg 4645 . . . 4  |-  ( F  e.  V  ->  dom  F  e.  _V )
6 inex1g 3935 . . . 4  |-  ( dom 
F  e.  _V  ->  ( dom  F  i^i  dom  G )  e.  _V )
7 mptexg 5439 . . . 4  |-  ( ( dom  F  i^i  dom  G )  e.  _V  ->  ( x  e.  ( dom 
F  i^i  dom  G ) 
|->  ( ( F `  x ) R ( G `  x ) ) )  e.  _V )
85, 6, 73syl 17 . . 3  |-  ( F  e.  V  ->  (
x  e.  ( dom 
F  i^i  dom  G ) 
|->  ( ( F `  x ) R ( G `  x ) ) )  e.  _V )
98adantr 270 . 2  |-  ( ( F  e.  V  /\  G  e.  W )  ->  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( ( F `
 x ) R ( G `  x
) ) )  e. 
_V )
10 dmeq 4584 . . . . 5  |-  ( a  =  F  ->  dom  a  =  dom  F )
11 dmeq 4584 . . . . 5  |-  ( b  =  G  ->  dom  b  =  dom  G )
1210, 11ineqan12d 3186 . . . 4  |-  ( ( a  =  F  /\  b  =  G )  ->  ( dom  a  i^i 
dom  b )  =  ( dom  F  i^i  dom 
G ) )
13 fveq1 5229 . . . . 5  |-  ( a  =  F  ->  (
a `  x )  =  ( F `  x ) )
14 fveq1 5229 . . . . 5  |-  ( b  =  G  ->  (
b `  x )  =  ( G `  x ) )
1513, 14oveqan12d 5583 . . . 4  |-  ( ( a  =  F  /\  b  =  G )  ->  ( ( a `  x ) R ( b `  x ) )  =  ( ( F `  x ) R ( G `  x ) ) )
1612, 15mpteq12dv 3881 . . 3  |-  ( ( a  =  F  /\  b  =  G )  ->  ( x  e.  ( dom  a  i^i  dom  b )  |->  ( ( a `  x ) R ( b `  x ) ) )  =  ( x  e.  ( dom  F  i^i  dom 
G )  |->  ( ( F `  x ) R ( G `  x ) ) ) )
17 df-of 5764 . . 3  |-  oF R  =  ( a  e.  _V ,  b  e.  _V  |->  ( x  e.  ( dom  a  i^i  dom  b )  |->  ( ( a `  x
) R ( b `
 x ) ) ) )
1816, 17ovmpt2ga 5682 . 2  |-  ( ( F  e.  _V  /\  G  e.  _V  /\  (
x  e.  ( dom 
F  i^i  dom  G ) 
|->  ( ( F `  x ) R ( G `  x ) ) )  e.  _V )  ->  ( F  oF R G )  =  ( x  e.  ( dom  F  i^i  dom 
G )  |->  ( ( F `  x ) R ( G `  x ) ) ) )
192, 4, 9, 18syl3anc 1170 1  |-  ( ( F  e.  V  /\  G  e.  W )  ->  ( F  oF R G )  =  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( ( F `
 x ) R ( G `  x
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434   _Vcvv 2610    i^i cin 2982    |-> cmpt 3860   dom cdm 4392   ` cfv 4953  (class class class)co 5564    oFcof 5762
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3914  ax-sep 3917  ax-pow 3969  ax-pr 3993  ax-un 4217  ax-setind 4309
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2826  df-csb 2919  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-iun 3701  df-br 3807  df-opab 3861  df-mpt 3862  df-id 4077  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-rn 4403  df-res 4404  df-ima 4405  df-iota 4918  df-fun 4955  df-fn 4956  df-f 4957  df-f1 4958  df-fo 4959  df-f1o 4960  df-fv 4961  df-ov 5567  df-oprab 5568  df-mpt2 5569  df-of 5764
This theorem is referenced by:  offres  5814
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