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Theorem offn 6319
Description: The function operation produces a function. (Contributed by Mario Carneiro, 22-Jul-2014.)
Hypotheses
Ref Expression
offval.1  |-  ( ph  ->  F  Fn  A )
offval.2  |-  ( ph  ->  G  Fn  B )
offval.3  |-  ( ph  ->  A  e.  V )
offval.4  |-  ( ph  ->  B  e.  W )
offval.5  |-  ( A  i^i  B )  =  S
Assertion
Ref Expression
offn  |-  ( ph  ->  ( F  o F R G )  Fn  S )

Proof of Theorem offn
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ovex 6109 . . 3  |-  ( ( F `  x ) R ( G `  x ) )  e. 
_V
2 eqid 2438 . . 3  |-  ( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) )  =  ( x  e.  S  |->  ( ( F `
 x ) R ( G `  x
) ) )
31, 2fnmpti 5576 . 2  |-  ( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) )  Fn  S
4 offval.1 . . . 4  |-  ( ph  ->  F  Fn  A )
5 offval.2 . . . 4  |-  ( ph  ->  G  Fn  B )
6 offval.3 . . . 4  |-  ( ph  ->  A  e.  V )
7 offval.4 . . . 4  |-  ( ph  ->  B  e.  W )
8 offval.5 . . . 4  |-  ( A  i^i  B )  =  S
9 eqidd 2439 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
10 eqidd 2439 . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  ( G `  x )  =  ( G `  x ) )
114, 5, 6, 7, 8, 9, 10offval 6315 . . 3  |-  ( ph  ->  ( F  o F R G )  =  ( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) ) )
1211fneq1d 5539 . 2  |-  ( ph  ->  ( ( F  o F R G )  Fn  S  <->  ( x  e.  S  |->  ( ( F `
 x ) R ( G `  x
) ) )  Fn  S ) )
133, 12mpbiri 226 1  |-  ( ph  ->  ( F  o F R G )  Fn  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    i^i cin 3321    e. cmpt 4269    Fn wfn 5452   ` cfv 5457  (class class class)co 6084    o Fcof 6306
This theorem is referenced by:  offveq  6328  ofsubeq0  10002  ofnegsub  10003  ofsubge0  10004  seqof  11385  psrbagcon  16441  i1faddlem  19588  i1fmullem  19589  dv11cn  19890  coemulc  20178  ofmulrt  20204  plydivlem3  20217  plyrem  20227  jensen  20832  basellem9  20876  frlmsslsp  27239  frlmup1  27241  caofcan  27531  ofmul12  27533  ofdivrec  27534  ofdivcan4  27535  ofdivdiv2  27536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-of 6308
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