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Theorem offn 6442
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  oF R G )  Fn  S )

Proof of Theorem offn
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ovex 6226 . . 3  |-  ( ( F `  x ) R ( G `  x ) )  e. 
_V
2 eqid 2454 . . 3  |-  ( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) )  =  ( x  e.  S  |->  ( ( F `
 x ) R ( G `  x
) ) )
31, 2fnmpti 5648 . 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 2455 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
10 eqidd 2455 . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  ( G `  x )  =  ( G `  x ) )
114, 5, 6, 7, 8, 9, 10offval 6438 . . 3  |-  ( ph  ->  ( F  oF R G )  =  ( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) ) )
1211fneq1d 5610 . 2  |-  ( ph  ->  ( ( F  oF R G )  Fn  S  <->  ( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) )  Fn  S ) )
133, 12mpbiri 233 1  |-  ( ph  ->  ( F  oF R G )  Fn  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    i^i cin 3436    |-> cmpt 4459    Fn wfn 5522   ` cfv 5527  (class class class)co 6201    oFcof 6429
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-of 6431
This theorem is referenced by:  offveq  6452  suppofss1d  6837  suppofss2d  6838  ofsubeq0  10431  ofnegsub  10432  ofsubge0  10433  seqof  11981  lcomfsupp  17109  psrbagcon  17564  psrbagev1  17719  frlmsslsp  18349  frlmsslspOLD  18350  frlmup1  18352  i1faddlem  21305  i1fmullem  21306  dv11cn  21607  coemulc  21856  ofmulrt  21882  plydivlem3  21895  plyrem  21905  jensen  22516  basellem9  22560  ofccat  27086  caofcan  29746  ofmul12  29748  ofdivrec  29749  ofdivcan4  29750  ofdivdiv2  29751  mndpsuppss  30933  mndpfsupp  30939
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