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Theorem fnfvrnss 6430
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 6428 . 2 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
2 frn 6091 . 2 (𝐹:𝐴𝐵 → ran 𝐹𝐵)
31, 2sylbir 225 1 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → ran 𝐹𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2030  wral 2941  wss 3607  ran crn 5144   Fn wfn 5921  wf 5922  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934
This theorem is referenced by:  ffvresb  6434  dffi3  8378  infxpenlem  8874  alephsing  9136  srgfcl  18561  mplind  19550  1stckgenlem  21404  psmetxrge0  22165  plyreres  24083  aannenlem1  24128  subuhgr  26223  subupgr  26224  subumgr  26225  subusgr  26226  rmulccn  30102  esumfsup  30260  sxbrsigalem3  30462  sitgf  30537  dihf11lem  36872  hdmaprnN  37473  hgmaprnN  37510  ntrrn  38737  volicoff  40530  dirkercncflem2  40639  fourierdlem15  40657  fourierdlem42  40684
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