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Theorem fnfvof 6106
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  o F R G ) `  X )  =  ( ( F `  X
) R ( G `
 X ) ) )

Proof of Theorem fnfvof
StepHypRef Expression
1 simpll 730 . . 3  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  A  e.  V )  ->  F  Fn  A )
2 simplr 731 . . 3  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  A  e.  V )  ->  G  Fn  A )
3 simpr 447 . . 3  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  A  e.  V )  ->  A  e.  V )
4 inidm 3391 . . 3  |-  ( A  i^i  A )  =  A
5 eqidd 2297 . . 3  |-  ( ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  A  e.  V )  /\  X  e.  A )  ->  ( F `  X )  =  ( F `  X ) )
6 eqidd 2297 . . 3  |-  ( ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  A  e.  V )  /\  X  e.  A )  ->  ( G `  X )  =  ( G `  X ) )
71, 2, 3, 3, 4, 5, 6ofval 6103 . 2  |-  ( ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  A  e.  V )  /\  X  e.  A )  ->  (
( F  o F R G ) `  X )  =  ( ( F `  X
) R ( G `
 X ) ) )
87anasss 628 1  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  ( A  e.  V  /\  X  e.  A ) )  -> 
( ( F  o F R G ) `  X )  =  ( ( F `  X
) R ( G `
 X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    Fn wfn 5266   ` cfv 5271  (class class class)co 5874    o Fcof 6092
This theorem is referenced by:  ghmplusg  15154  lmhmplusg  15817  coe1addfv  16358  nmotri  18264  evlslem3  19414  evlslem1  19415  evl1addd  19433  evl1subd  19434  evl1muld  19435  mdegaddle  19476  ply1rem  19565  fta1glem2  19568  fta1blem  19570  plyexmo  19709  ulmdvlem1  19793  jensen  20299  dchrmulcl  20504  dchrinv  20516  sumdchr2  20525  dchr2sum  20528  lcomfsup  26871  mzpsubst  26929  mzpcong  27162  frlmvscaval  27334  frlmsslsp  27351  frlmup1  27353  frlmup2  27354  islindf4  27411  rngunsnply  27481  mamudi  27564  mamudir  27565
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-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
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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