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Theorem fnfvrnss 7121
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 7119 . 2 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
2 frn 6723 . 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 394  wcel 2104  wral 3059  wss 3947  ran crn 5676   Fn wfn 6537  wf 6538  cfv 6542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550
This theorem is referenced by:  ffvresb  7125  dffi3  9428  infxpenlem  10010  alephsing  10273  seqexw  13986  srgfcl  20090  mplind  21850  1stckgenlem  23277  psmetxrge0  24039  plyreres  26032  aannenlem1  26077  subuhgr  28810  subupgr  28811  subumgr  28812  subusgr  28813  elrspunidl  32820  rmulccn  33206  esumfsup  33366  sxbrsigalem3  33569  sitgf  33644  gg-rmulccn  35465  ctbssinf  36590  dihf11lem  40440  hdmaprnN  41038  hgmaprnN  41075  ofoafg  42406  naddcnff  42414  ntrrn  43175  mnurndlem1  43342  volicoff  45009  dirkercncflem2  45118  fourierdlem15  45136  fourierdlem42  45163
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