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Theorem offn 6326
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 6111 . . 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 5534 . 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 6322 . . 3  |-  ( ph  ->  ( F  oF R G )  =  ( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) ) )
1211fneq1d 5496 . 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 1369    e. wcel 1756    i^i cin 3322    e. cmpt 4345    Fn wfn 5408   ` cfv 5413  (class class class)co 6086    oFcof 6313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pr 4526
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315
This theorem is referenced by:  offveq  6336  suppofss1d  6721  suppofss2d  6722  ofsubeq0  10311  ofnegsub  10312  ofsubge0  10313  seqof  11855  lcomfsupp  16963  psrbagcon  17417  psrbagev1  17569  frlmsslsp  18198  frlmsslspOLD  18199  frlmup1  18201  i1faddlem  21146  i1fmullem  21147  dv11cn  21448  coemulc  21697  ofmulrt  21723  plydivlem3  21736  plyrem  21746  jensen  22357  basellem9  22401  ofccat  26893  caofcan  29550  ofmul12  29552  ofdivrec  29553  ofdivcan4  29554  ofdivdiv2  29555  mndpsuppss  30735  mndpfsupp  30740
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