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Theorem fnfvrnss 5377
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 5375 . 2  |-  ( F : A --> B  <->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B
) )
2 frn 5103 . 2  |-  ( F : A --> B  ->  ran  F  C_  B )
31, 2sylbir 133 1  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B )  ->  ran  F  C_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    e. wcel 1434   A.wral 2353    C_ wss 2982   ran crn 4392    Fn wfn 4947   -->wf 4948   ` cfv 4952
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-sbc 2825  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-mpt 3861  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-fv 4960
This theorem is referenced by:  ffvresb  5380
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