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Theorem fnfvrnss 7119
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 7117 . 2 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
2 frn 6724 . 2 (𝐹:𝐴𝐵 → ran 𝐹𝐵)
31, 2sylbir 234 1 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → ran 𝐹𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2106  wral 3061  wss 3948  ran crn 5677   Fn wfn 6538  wf 6539  cfv 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551
This theorem is referenced by:  ffvresb  7123  dffi3  9425  infxpenlem  10007  alephsing  10270  seqexw  13981  srgfcl  20018  mplind  21630  1stckgenlem  23056  psmetxrge0  23818  plyreres  25795  aannenlem1  25840  subuhgr  28540  subupgr  28541  subumgr  28542  subusgr  28543  elrspunidl  32541  rmulccn  32903  esumfsup  33063  sxbrsigalem3  33266  sitgf  33341  gg-rmulccn  35174  ctbssinf  36282  dihf11lem  40132  hdmaprnN  40730  hgmaprnN  40767  ofoafg  42094  naddcnff  42102  ntrrn  42863  mnurndlem1  43030  volicoff  44701  dirkercncflem2  44810  fourierdlem15  44828  fourierdlem42  44855
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