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Theorem offval3 6107
Description: General value of  ( F  o F 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  o F 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 2809 . . 3  |-  ( F  e.  V  ->  F  e.  _V )
21adantr 451 . 2  |-  ( ( F  e.  V  /\  G  e.  W )  ->  F  e.  _V )
3 elex 2809 . . 3  |-  ( G  e.  W  ->  G  e.  _V )
43adantl 452 . 2  |-  ( ( F  e.  V  /\  G  e.  W )  ->  G  e.  _V )
5 dmexg 4955 . . . 4  |-  ( F  e.  V  ->  dom  F  e.  _V )
6 inex1g 4173 . . . 4  |-  ( dom 
F  e.  _V  ->  ( dom  F  i^i  dom  G )  e.  _V )
7 mptexg 5761 . . . 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 18 . . 3  |-  ( F  e.  V  ->  (
x  e.  ( dom 
F  i^i  dom  G ) 
|->  ( ( F `  x ) R ( G `  x ) ) )  e.  _V )
98adantr 451 . 2  |-  ( ( F  e.  V  /\  G  e.  W )  ->  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( ( F `
 x ) R ( G `  x
) ) )  e. 
_V )
10 dmeq 4895 . . . . 5  |-  ( a  =  F  ->  dom  a  =  dom  F )
11 dmeq 4895 . . . . 5  |-  ( b  =  G  ->  dom  b  =  dom  G )
1210, 11ineqan12d 3385 . . . 4  |-  ( ( a  =  F  /\  b  =  G )  ->  ( dom  a  i^i 
dom  b )  =  ( dom  F  i^i  dom 
G ) )
13 fveq1 5540 . . . . 5  |-  ( a  =  F  ->  (
a `  x )  =  ( F `  x ) )
14 fveq1 5540 . . . . 5  |-  ( b  =  G  ->  (
b `  x )  =  ( G `  x ) )
1513, 14oveqan12d 5893 . . . 4  |-  ( ( a  =  F  /\  b  =  G )  ->  ( ( a `  x ) R ( b `  x ) )  =  ( ( F `  x ) R ( G `  x ) ) )
1612, 15mpteq12dv 4114 . . 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 6094 . . 3  |-  o F R  =  ( a  e.  _V ,  b  e.  _V  |->  ( x  e.  ( dom  a  i^i  dom  b )  |->  ( ( a `  x
) R ( b `
 x ) ) ) )
1816, 17ovmpt2ga 5993 . 2  |-  ( ( F  e.  _V  /\  G  e.  _V  /\  (
x  e.  ( dom 
F  i^i  dom  G ) 
|->  ( ( F `  x ) R ( G `  x ) ) )  e.  _V )  ->  ( F  o F R G )  =  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( ( F `
 x ) R ( G `  x
) ) ) )
192, 4, 9, 18syl3anc 1182 1  |-  ( ( F  e.  V  /\  G  e.  W )  ->  ( F  o F 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 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    i^i cin 3164    e. cmpt 4093   dom cdm 4705   ` cfv 5271  (class class class)co 5874    o Fcof 6092
This theorem is referenced by:  offres  6108
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-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
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-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  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-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094
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