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Theorem offval3 6318
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 2964 . . 3  |-  ( F  e.  V  ->  F  e.  _V )
21adantr 452 . 2  |-  ( ( F  e.  V  /\  G  e.  W )  ->  F  e.  _V )
3 elex 2964 . . 3  |-  ( G  e.  W  ->  G  e.  _V )
43adantl 453 . 2  |-  ( ( F  e.  V  /\  G  e.  W )  ->  G  e.  _V )
5 dmexg 5130 . . . 4  |-  ( F  e.  V  ->  dom  F  e.  _V )
6 inex1g 4346 . . . 4  |-  ( dom 
F  e.  _V  ->  ( dom  F  i^i  dom  G )  e.  _V )
7 mptexg 5965 . . . 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 19 . . 3  |-  ( F  e.  V  ->  (
x  e.  ( dom 
F  i^i  dom  G ) 
|->  ( ( F `  x ) R ( G `  x ) ) )  e.  _V )
98adantr 452 . 2  |-  ( ( F  e.  V  /\  G  e.  W )  ->  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( ( F `
 x ) R ( G `  x
) ) )  e. 
_V )
10 dmeq 5070 . . . . 5  |-  ( a  =  F  ->  dom  a  =  dom  F )
11 dmeq 5070 . . . . 5  |-  ( b  =  G  ->  dom  b  =  dom  G )
1210, 11ineqan12d 3544 . . . 4  |-  ( ( a  =  F  /\  b  =  G )  ->  ( dom  a  i^i 
dom  b )  =  ( dom  F  i^i  dom 
G ) )
13 fveq1 5727 . . . . 5  |-  ( a  =  F  ->  (
a `  x )  =  ( F `  x ) )
14 fveq1 5727 . . . . 5  |-  ( b  =  G  ->  (
b `  x )  =  ( G `  x ) )
1513, 14oveqan12d 6100 . . . 4  |-  ( ( a  =  F  /\  b  =  G )  ->  ( ( a `  x ) R ( b `  x ) )  =  ( ( F `  x ) R ( G `  x ) ) )
1612, 15mpteq12dv 4287 . . 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 6305 . . 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 6203 . 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 1184 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 359    = wceq 1652    e. wcel 1725   _Vcvv 2956    i^i cin 3319    e. cmpt 4266   dom cdm 4878   ` cfv 5454  (class class class)co 6081    o Fcof 6303
This theorem is referenced by:  offres  6319
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305
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