MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fnfvrnss Structured version   Visualization version   GIF version

Theorem fnfvrnss 6882
Description: An upper bound for range determined by function values. (Contributed by NM, 8-Oct-2004.)
Assertion
Ref Expression
fnfvrnss ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → ran 𝐹𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹

Proof of Theorem fnfvrnss
StepHypRef Expression
1 ffnfv 6880 . 2 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
2 frn 6519 . 2 (𝐹:𝐴𝐵 → ran 𝐹𝐵)
31, 2sylbir 236 1 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → ran 𝐹𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2107  wral 3143  wss 3940  ran crn 5555   Fn wfn 6349  wf 6350  cfv 6354
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-fv 6362
This theorem is referenced by:  ffvresb  6886  dffi3  8889  infxpenlem  9433  alephsing  9692  seqexw  13380  srgfcl  19201  mplind  20217  1stckgenlem  22096  psmetxrge0  22857  plyreres  24806  aannenlem1  24851  subuhgr  27001  subupgr  27002  subumgr  27003  subusgr  27004  rmulccn  31076  esumfsup  31234  sxbrsigalem3  31435  sitgf  31510  ctbssinf  34575  dihf11lem  38288  hdmaprnN  38886  hgmaprnN  38923  ntrrn  40356  mnurndlem1  40501  volicoff  42165  dirkercncflem2  42274  fourierdlem15  42292  fourierdlem42  42319
  Copyright terms: Public domain W3C validator