MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fnfvof Structured version   Unicode version

Theorem fnfvof 6322
Description: Function value of a pointwise composition. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Proof shortened by Mario Carneiro, 5-Jun-2015.)
Assertion
Ref Expression
fnfvof  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  ( A  e.  V  /\  X  e.  A ) )  -> 
( ( F  oF R G ) `
 X )  =  ( ( F `  X ) R ( G `  X ) ) )

Proof of Theorem fnfvof
StepHypRef Expression
1 simpll 746 . . 3  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  A  e.  V )  ->  F  Fn  A )
2 simplr 747 . . 3  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  A  e.  V )  ->  G  Fn  A )
3 simpr 458 . . 3  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  A  e.  V )  ->  A  e.  V )
4 inidm 3547 . . 3  |-  ( A  i^i  A )  =  A
5 eqidd 2434 . . 3  |-  ( ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  A  e.  V )  /\  X  e.  A )  ->  ( F `  X )  =  ( F `  X ) )
6 eqidd 2434 . . 3  |-  ( ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  A  e.  V )  /\  X  e.  A )  ->  ( G `  X )  =  ( G `  X ) )
71, 2, 3, 3, 4, 5, 6ofval 6318 . 2  |-  ( ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  A  e.  V )  /\  X  e.  A )  ->  (
( F  oF R G ) `  X )  =  ( ( F `  X
) R ( G `
 X ) ) )
87anasss 640 1  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  ( A  e.  V  /\  X  e.  A ) )  -> 
( ( F  oF R G ) `
 X )  =  ( ( F `  X ) R ( G `  X ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1362    e. wcel 1755    Fn wfn 5401   ` cfv 5406  (class class class)co 6080    oFcof 6307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pr 4519
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309
This theorem is referenced by:  ghmplusg  16308  lcomfsupOLD  16908  lcomfsupp  16909  lmhmplusg  17047  coe1addfv  17617  frlmvscaval  18036  frlmsslsp  18065  frlmsslspOLD  18066  frlmup1  18068  frlmup2  18069  islindf4  18109  mamudi  18149  mamudir  18150  mdetrlin  18251  nmotri  20160  evlslem3  21366  evlslem1  21367  evl1addd  21385  evl1subd  21386  evl1muld  21387  mdegaddle  21430  ply1rem  21520  fta1glem2  21523  fta1blem  21525  plyexmo  21664  ulmdvlem1  21750  jensen  22267  dchrmulcl  22473  dchrinv  22485  sumdchr2  22494  dchr2sum  22497  ofccat  26789  mzpsubst  28930  mzpcong  29160  rngunsnply  29375  lincsum  30693
  Copyright terms: Public domain W3C validator