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Theorem fnfvrnss 7104
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 7102 . 2 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
2 frn 6711 . 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 3060  wss 3944  ran crn 5670   Fn wfn 6527  wf 6528  cfv 6532
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 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-fv 6540
This theorem is referenced by:  ffvresb  7108  dffi3  9408  infxpenlem  9990  alephsing  10253  seqexw  13964  srgfcl  19977  mplind  21560  1stckgenlem  22986  psmetxrge0  23748  plyreres  25725  aannenlem1  25770  subuhgr  28408  subupgr  28409  subumgr  28410  subusgr  28411  elrspunidl  32397  rmulccn  32739  esumfsup  32899  sxbrsigalem3  33102  sitgf  33177  ctbssinf  36091  dihf11lem  39942  hdmaprnN  40540  hgmaprnN  40577  ofoafg  41875  naddcnff  41883  ntrrn  42644  mnurndlem1  42811  volicoff  44484  dirkercncflem2  44593  fourierdlem15  44611  fourierdlem42  44638
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