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Theorem offval3 6091
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 2796 . . 3  |-  ( F  e.  V  ->  F  e.  _V )
21adantr 451 . 2  |-  ( ( F  e.  V  /\  G  e.  W )  ->  F  e.  _V )
3 elex 2796 . . 3  |-  ( G  e.  W  ->  G  e.  _V )
43adantl 452 . 2  |-  ( ( F  e.  V  /\  G  e.  W )  ->  G  e.  _V )
5 dmexg 4939 . . . 4  |-  ( F  e.  V  ->  dom  F  e.  _V )
6 inex1g 4157 . . . 4  |-  ( dom 
F  e.  _V  ->  ( dom  F  i^i  dom  G )  e.  _V )
7 mptexg 5745 . . . 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 4879 . . . . 5  |-  ( a  =  F  ->  dom  a  =  dom  F )
11 dmeq 4879 . . . . 5  |-  ( b  =  G  ->  dom  b  =  dom  G )
1210, 11ineqan12d 3372 . . . 4  |-  ( ( a  =  F  /\  b  =  G )  ->  ( dom  a  i^i 
dom  b )  =  ( dom  F  i^i  dom 
G ) )
13 fveq1 5524 . . . . 5  |-  ( a  =  F  ->  (
a `  x )  =  ( F `  x ) )
14 fveq1 5524 . . . . 5  |-  ( b  =  G  ->  (
b `  x )  =  ( G `  x ) )
1513, 14oveqan12d 5877 . . . 4  |-  ( ( a  =  F  /\  b  =  G )  ->  ( ( a `  x ) R ( b `  x ) )  =  ( ( F `  x ) R ( G `  x ) ) )
1612, 15mpteq12dv 4098 . . 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 6078 . . 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 5977 . 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 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151    e. cmpt 4077   dom cdm 4689   ` cfv 5255  (class class class)co 5858    o Fcof 6076
This theorem is referenced by:  offres  6092
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078
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