Theorem fnfvrnss 7093

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 7091	. 2 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵))
2		frn 6695	. 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 2109  ∀wral 3044   ⊆ wss 3914  ran crn 5639   Fn wfn 6506  ⟶wf 6507  ‘cfv 6511

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fv 6519

This theorem is referenced by:  ffvresb  7097  dffi3  9382  infxpenlem  9966  alephsing  10229  seqexw  13982  srgfcl  20105  mplind  21977  1stckgenlem  23440  psmetxrge0  24201  plyreres  26190  aannenlem1  26236  bdayn0sf1o  28259  dfnns2  28261  subuhgr  29213  subupgr  29214  subumgr  29215  subusgr  29216  elrspunidl  33399  rmulccn  33918  esumfsup  34060  sxbrsigalem3  34263  sitgf  34338  ctbssinf  37394  dihf11lem  41260  hdmaprnN  41858  hgmaprnN  41895  ofoafg  43343  naddcnff  43351  ntrrn  44111  mnurndlem1  44270  volicoff  45993  dirkercncflem2  46102  fourierdlem15  46120  fourierdlem42  46147  grimuhgr  47887  slotresfo  48887