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

Theorem fsuppimp 7826
 Description: Implications of a class being a finitely supported function (in relation to a given zero). (Contributed by AV, 26-May-2019.)
Assertion
Ref Expression
fsuppimp finSupp supp

Proof of Theorem fsuppimp
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-fsupp 7821 . . . . 5 finSupp supp
21relopabi 5121 . . . 4 finSupp
32brrelexi 5034 . . 3 finSupp
42brrelex2i 5035 . . 3 finSupp
53, 4jca 532 . 2 finSupp
6 isfsupp 7824 . . 3 finSupp supp
76biimpd 207 . 2 finSupp supp
85, 7mpcom 36 1 finSupp supp
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   wcel 1762  cvv 3108   class class class wbr 4442   wfun 5575  (class class class)co 6277   supp csupp 6893  cfn 7508   finSupp cfsupp 7820 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-iota 5544  df-fun 5583  df-fv 5589  df-ov 6280  df-fsupp 7821 This theorem is referenced by:  fsuppimpd  7827  fsuppunfi  7840  fsuppunbi  7841  fsuppres  7845  resfsupp  7847  fsuppco  7852  oemapvali  8094  mptnn0fsuppr  12063  gsumzres  16700  gsumzf1o  16703  gsumzadd  16721  mdegldg  22196
 Copyright terms: Public domain W3C validator