Theorem fnfvrnss 5763

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 5761	. 2 |- ( F : A --> B <-> ( F Fn A /\ A. x e. A ( F ` x ) e. B ) )
2		frn 5454	. 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 2178   A.wral 2486   C_ wss 3174   ran crn 4694   Fn wfn 5285   -->wf 5286   ` cfv 5290

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-fv 5298

This theorem is referenced by:  ffvresb  5766  inresflem  7188  srgfcl  13850  psmetxrge0  14919  plyreres  15351