MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fnfvrnss Structured version   Unicode version

Theorem fnfvrnss 6040
Description: An upper bound for range determined by function values. (Contributed by NM, 8-Oct-2004.)
Assertion
Ref Expression
fnfvrnss  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B )  ->  ran  F  C_  B )
Distinct variable groups:    x, A    x, B    x, F

Proof of Theorem fnfvrnss
StepHypRef Expression
1 ffnfv 6038 . 2  |-  ( F : A --> B  <->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B
) )
2 frn 5728 . 2  |-  ( F : A --> B  ->  ran  F  C_  B )
31, 2sylbir 213 1  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B )  ->  ran  F  C_  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1762   A.wral 2807    C_ wss 3469   ran crn 4993    Fn wfn 5574   -->wf 5575   ` cfv 5579
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-fv 5587
This theorem is referenced by:  ffvresb  6043  dffi3  7880  infxpenlem  8380  alephsing  8645  mplind  17931  1stckgenlem  19782  psmetxrge0  20545  plyreres  22406  aannenlem1  22451  rmulccn  27396  esumfsup  27566  sxbrsigalem3  27733  sitgf  27779  dirkercncflem2  31223  fourierdlem15  31241  fourierdlem42  31268  dihf11lem  35938  hdmaprnN  36539  hgmaprnN  36576
  Copyright terms: Public domain W3C validator