MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  offn Unicode version

Theorem offn 6301
Description: The function operation produces a function. (Contributed by Mario Carneiro, 22-Jul-2014.)
Hypotheses
Ref Expression
offval.1
offval.2
offval.3
offval.4
offval.5
Assertion
Ref Expression
offn

Proof of Theorem offn
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ovex 6086 . . 3
2 eqid 2422 . . 3
31, 2fnmpti 5509 . 2
4 offval.1 . . . 4
5 offval.2 . . . 4
6 offval.3 . . . 4
7 offval.4 . . . 4
8 offval.5 . . . 4
9 eqidd 2423 . . . 4
10 eqidd 2423 . . . 4
114, 5, 6, 7, 8, 9, 10offval 6297 . . 3
1211fneq1d 5471 . 2
133, 12mpbiri 227 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  /\wa 362  =wceq 1687  e.wcel 1749  i^icin 3304  e.cmpt 4325  Fnwfn 5385  `cfv 5390  (class class class)co 6061  oFcof 6288
This theorem is referenced by:  offveq  6311  suppofss1d  6690  suppofss2d  6691  ofsubeq0  10265  ofnegsub  10266  ofsubge0  10267  seqof  11804  psrbagcon  17256  frlmsslsp  17924  frlmup1  17926  i1faddlem  20871  i1fmullem  20872  dv11cn  21173  coemulc  21463  ofmulrt  21489  plydivlem3  21502  plyrem  21512  jensen  22123  basellem9  22167  ofccat  26644  caofcan  29270  ofmul12  29272  ofdivrec  29273  ofdivcan4  29274  ofdivdiv2  29275  mndpsuppss  30448  mndpfsupp  30457  frlmXsslsp  30520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1586  ax-4 1597  ax-5 1661  ax-6 1701  ax-7 1721  ax-9 1753  ax-10 1768  ax-11 1773  ax-12 1785  ax-13 1934  ax-ext 2403  ax-rep 4378  ax-sep 4388  ax-nul 4396  ax-pr 4503
This theorem depends on definitions:  df-bi 179  df-or 363  df-an 364  df-3an 952  df-tru 1355  df-ex 1582  df-nf 1585  df-sb 1694  df-eu 2248  df-mo 2249  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2547  df-ne 2587  df-ral 2699  df-rex 2700  df-reu 2701  df-rab 2703  df-v 2953  df-sbc 3165  df-csb 3266  df-dif 3308  df-un 3310  df-in 3312  df-ss 3319  df-nul 3615  df-if 3769  df-sn 3859  df-pr 3860  df-op 3862  df-uni 4067  df-iun 4148  df-br 4268  df-opab 4326  df-mpt 4327  df-id 4607  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5353  df-fun 5392  df-fn 5393  df-f 5394  df-f1 5395  df-fo 5396  df-f1o 5397  df-fv 5398  df-ov 6064  df-oprab 6065  df-mpt2 6066  df-of 6290
  Copyright terms: Public domain W3C validator