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Theorem offn 6320
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 6105 . . 3  |-  ( ( F `  x ) R ( G `  x ) )  e. 
_V
2 eqid 2433 . . 3  |-  ( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) )  =  ( x  e.  S  |->  ( ( F `
 x ) R ( G `  x
) ) )
31, 2fnmpti 5527 . 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 2434 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
10 eqidd 2434 . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  ( G `  x )  =  ( G `  x ) )
114, 5, 6, 7, 8, 9, 10offval 6316 . . 3  |-  ( ph  ->  ( F  oF R G )  =  ( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) ) )
1211fneq1d 5489 . 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 1362    e. wcel 1755    i^i cin 3315    e. cmpt 4338    Fn wfn 5401   ` cfv 5406  (class class class)co 6080    oFcof 6307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pr 4519
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309
This theorem is referenced by:  offveq  6330  suppofss1d  6715  suppofss2d  6716  ofsubeq0  10307  ofnegsub  10308  ofsubge0  10309  seqof  11847  lcomfsupp  16909  psrbagcon  17374  psrbagev1  17522  frlmsslsp  18065  frlmsslspOLD  18066  frlmup1  18068  i1faddlem  21013  i1fmullem  21014  dv11cn  21315  coemulc  21607  ofmulrt  21633  plydivlem3  21646  plyrem  21656  jensen  22267  basellem9  22311  ofccat  26789  caofcan  29442  ofmul12  29444  ofdivrec  29445  ofdivcan4  29446  ofdivdiv2  29447  mndpsuppss  30616  mndpfsupp  30621
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