Theorem fnfvrnss 5696

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

Step	Hyp	Ref	Expression
1		ffnfv 5694	. 2 |- ( F : A --> B <-> ( F Fn A /\ A. x e. A ( F ` x ) e. B ) )
2		frn 5393	. 2 |- ( F : A --> B -> ran F C_ B )
3	1, 2	sylbir 135	1 |- ( ( F Fn A /\ A. x e. A ( F ` x ) e. B ) -> ran F C_ B )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2160   A.wral 2468   C_ wss 3144   ran crn 4645   Fn wfn 5230   -->wf 5231   ` cfv 5235

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fv 5243

This theorem is referenced by:  ffvresb  5699  inresflem  7088  srgfcl  13324  psmetxrge0  14284