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

Theorem fnfvrnss 7141
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 7139 . 2 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
2 frn 6744 . 2 (𝐹:𝐴𝐵 → ran 𝐹𝐵)
31, 2sylbir 235 1 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → ran 𝐹𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2106  wral 3059  wss 3963  ran crn 5690   Fn wfn 6558  wf 6559  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571
This theorem is referenced by:  ffvresb  7145  dffi3  9469  infxpenlem  10051  alephsing  10314  seqexw  14055  srgfcl  20214  mplind  22112  1stckgenlem  23577  psmetxrge0  24339  plyreres  26339  aannenlem1  26385  dfnns2  28377  subuhgr  29318  subupgr  29319  subumgr  29320  subusgr  29321  elrspunidl  33436  rmulccn  33889  esumfsup  34051  sxbrsigalem3  34254  sitgf  34329  ctbssinf  37389  dihf11lem  41249  hdmaprnN  41847  hgmaprnN  41884  ofoafg  43344  naddcnff  43352  ntrrn  44112  mnurndlem1  44277  volicoff  45951  dirkercncflem2  46060  fourierdlem15  46078  fourierdlem42  46105  grimuhgr  47816
  Copyright terms: Public domain W3C validator