Theorem fnfvrnss 7063

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 7061	. 2 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵))
2		frn 6666	. 2 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵)
3	1, 2	sylbir 235	1 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) → ran 𝐹 ⊆ 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2113  ∀wral 3048   ⊆ wss 3898  ran crn 5622   Fn wfn 6484  ⟶wf 6485  ‘cfv 6489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-fv 6497

This theorem is referenced by:  ffvresb  7067  dffi3  9326  infxpenlem  9915  alephsing  10178  seqexw  13931  srgfcl  20122  mplind  22016  1stckgenlem  23488  psmetxrge0  24248  plyreres  26237  aannenlem1  26283  bdayn0sf1o  28315  dfnns2  28317  subuhgr  29285  subupgr  29286  subumgr  29287  subusgr  29288  elrspunidl  33437  rmulccn  34013  esumfsup  34155  sxbrsigalem3  34357  sitgf  34432  ctbssinf  37523  dihf11lem  41438  hdmaprnN  42036  hgmaprnN  42073  ofoafg  43511  naddcnff  43519  ntrrn  44279  mnurndlem1  44438  volicoff  46155  dirkercncflem2  46264  fourierdlem15  46282  fourierdlem42  46309  grimuhgr  48049  slotresfo  49060