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Theorem fnfvrnss 7072
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 7070 . 2 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
2 frn 6679 . 2 (𝐹:𝐴𝐵 → ran 𝐹𝐵)
31, 2sylbir 234 1 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → ran 𝐹𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2107  wral 3061  wss 3914  ran crn 5638   Fn wfn 6495  wf 6496  cfv 6500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508
This theorem is referenced by:  ffvresb  7076  dffi3  9375  infxpenlem  9957  alephsing  10220  seqexw  13931  srgfcl  19935  mplind  21501  1stckgenlem  22927  psmetxrge0  23689  plyreres  25666  aannenlem1  25711  subuhgr  28283  subupgr  28284  subumgr  28285  subusgr  28286  elrspunidl  32258  rmulccn  32573  esumfsup  32733  sxbrsigalem3  32936  sitgf  33011  ctbssinf  35927  dihf11lem  39779  hdmaprnN  40377  hgmaprnN  40414  ofoafg  41717  naddcnff  41725  ntrrn  42486  mnurndlem1  42653  volicoff  44326  dirkercncflem2  44435  fourierdlem15  44453  fourierdlem42  44480
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