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Theorem offn 6524
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 6298 . . 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 5691 . 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 6520 . . 3  |-  ( ph  ->  ( F  oF R G )  =  ( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) ) )
1211fneq1d 5653 . 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 367    = wceq 1398    e. wcel 1823    i^i cin 3460    |-> cmpt 4497    Fn wfn 5565   ` cfv 5570  (class class class)co 6270    oFcof 6511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513
This theorem is referenced by:  offveq  6534  suppofss1d  6929  suppofss2d  6930  ofsubeq0  10528  ofnegsub  10529  ofsubge0  10530  seqof  12146  lcomfsupp  17745  psrbagcon  18217  psrbagev1  18372  frlmsslsp  18998  frlmup1  19000  i1faddlem  22266  i1fmullem  22267  dv11cn  22568  coemulc  22818  ofmulrt  22844  plydivlem3  22857  plyrem  22867  jensen  23516  basellem9  23560  ofccat  28761  caofcan  31469  ofmul12  31471  ofdivrec  31472  ofdivcan4  31473  ofdivdiv2  31474  mndpsuppss  33218  mndpfsupp  33223
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