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Theorem offval3 5943
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 2644 . . 3  |-  ( F  e.  V  ->  F  e.  _V )
21adantr 271 . 2  |-  ( ( F  e.  V  /\  G  e.  W )  ->  F  e.  _V )
3 elex 2644 . . 3  |-  ( G  e.  W  ->  G  e.  _V )
43adantl 272 . 2  |-  ( ( F  e.  V  /\  G  e.  W )  ->  G  e.  _V )
5 dmexg 4729 . . . 4  |-  ( F  e.  V  ->  dom  F  e.  _V )
6 inex1g 3996 . . . 4  |-  ( dom 
F  e.  _V  ->  ( dom  F  i^i  dom  G )  e.  _V )
7 mptexg 5561 . . . 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 271 . 2  |-  ( ( F  e.  V  /\  G  e.  W )  ->  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( ( F `
 x ) R ( G `  x
) ) )  e. 
_V )
10 dmeq 4667 . . . . 5  |-  ( a  =  F  ->  dom  a  =  dom  F )
11 dmeq 4667 . . . . 5  |-  ( b  =  G  ->  dom  b  =  dom  G )
1210, 11ineqan12d 3218 . . . 4  |-  ( ( a  =  F  /\  b  =  G )  ->  ( dom  a  i^i 
dom  b )  =  ( dom  F  i^i  dom 
G ) )
13 fveq1 5339 . . . . 5  |-  ( a  =  F  ->  (
a `  x )  =  ( F `  x ) )
14 fveq1 5339 . . . . 5  |-  ( b  =  G  ->  (
b `  x )  =  ( G `  x ) )
1513, 14oveqan12d 5709 . . . 4  |-  ( ( a  =  F  /\  b  =  G )  ->  ( ( a `  x ) R ( b `  x ) )  =  ( ( F `  x ) R ( G `  x ) ) )
1612, 15mpteq12dv 3942 . . 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 5894 . . 3  |-  oF R  =  ( a  e.  _V ,  b  e.  _V  |->  ( x  e.  ( dom  a  i^i  dom  b )  |->  ( ( a `  x
) R ( b `
 x ) ) ) )
1816, 17ovmpt2ga 5812 . 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 1181 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 103    = wceq 1296    e. wcel 1445   _Vcvv 2633    i^i cin 3012    |-> cmpt 3921   dom cdm 4467   ` cfv 5049  (class class class)co 5690    oFcof 5892
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-of 5894
This theorem is referenced by:  offres  5944
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