Users' Mathboxes Mathbox for Steven Nguyen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fsuppfund Structured version   Visualization version   GIF version

Theorem fsuppfund 41522
Description: A finitely supported function is a function. (Contributed by SN, 8-Mar-2025.)
Hypothesis
Ref Expression
fsuppfund.1 (𝜑𝐹 finSupp 𝑍)
Assertion
Ref Expression
fsuppfund (𝜑 → Fun 𝐹)

Proof of Theorem fsuppfund
StepHypRef Expression
1 fsuppfund.1 . 2 (𝜑𝐹 finSupp 𝑍)
2 fsuppimp 9363 . . 3 (𝐹 finSupp 𝑍 → (Fun 𝐹 ∧ (𝐹 supp 𝑍) ∈ Fin))
32simpld 494 . 2 (𝐹 finSupp 𝑍 → Fun 𝐹)
41, 3syl 17 1 (𝜑 → Fun 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098   class class class wbr 5138  Fun wfun 6527  (class class class)co 7401   supp csupp 8140  Fincfn 8934   finSupp cfsupp 9356
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-iota 6485  df-fun 6535  df-fv 6541  df-ov 7404  df-fsupp 9357
This theorem is referenced by:  fsuppss  41524
  Copyright terms: Public domain W3C validator