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Theorem fnfvrnss 6530
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 6528 . 2 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
2 frn 6191 . 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 382  wcel 2145  wral 3061  wss 3723  ran crn 5250   Fn wfn 6024  wf 6025  cfv 6029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-fv 6037
This theorem is referenced by:  ffvresb  6534  dffi3  8491  infxpenlem  9034  alephsing  9298  srgfcl  18716  mplind  19710  1stckgenlem  21570  psmetxrge0  22331  plyreres  24251  aannenlem1  24296  subuhgr  26394  subupgr  26395  subumgr  26396  subusgr  26397  rmulccn  30307  esumfsup  30465  sxbrsigalem3  30667  sitgf  30742  dihf11lem  37069  hdmaprnN  37667  hgmaprnN  37704  ntrrn  38939  volicoff  40722  dirkercncflem2  40831  fourierdlem15  40849  fourierdlem42  40876
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