Theorem fnfvrnss 6883

Description: An upper bound for range determined by function values. (Contributed by NM, 8-Oct-2004.)

Assertion

Ref	Expression
fnfvrnss	⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) → ran 𝐹 ⊆ 𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹

Proof of Theorem fnfvrnss

Step	Hyp	Ref	Expression
1		ffnfv 6881	. 2 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵))
2		frn 6519	. 2 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵)
3	1, 2	sylbir 237	1 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) → ran 𝐹 ⊆ 𝐵)

Syntax hints:   → wi 4   ∧ wa 398   ∈ wcel 2110  ∀wral 3138   ⊆ wss 3935  ran crn 5555   Fn wfn 6349  ⟶wf 6350  ‘cfv 6354

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-fv 6362

This theorem is referenced by:  ffvresb  6887  dffi3  8894  infxpenlem  9438  alephsing  9697  seqexw  13384  srgfcl  19264  mplind  20281  1stckgenlem  22160  psmetxrge0  22922  plyreres  24871  aannenlem1  24916  subuhgr  27067  subupgr  27068  subumgr  27069  subusgr  27070  rmulccn  31171  esumfsup  31329  sxbrsigalem3  31530  sitgf  31605  ctbssinf  34686  dihf11lem  38401  hdmaprnN  38999  hgmaprnN  39036  ntrrn  40470  mnurndlem1  40615  volicoff  42279  dirkercncflem2  42388  fourierdlem15  42406  fourierdlem42  42433