Theorem fnima 5249

Description: The image of a function's domain is its range. (Contributed by NM, 4-Nov-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Assertion

Ref	Expression
fnima	|- ( F Fn A -> ( F " A ) = ran F )

Proof of Theorem fnima

Step	Hyp	Ref	Expression
1		df-ima 4560	. 2 |- ( F " A ) = ran ( F |` A )
2		fnresdm 5240	. . 3 |- ( F Fn A -> ( F |` A ) = F )
3	2	rneqd 4776	. 2 |- ( F Fn A -> ran ( F |` A ) = ran F )
4	1, 3	syl5eq 2185	1 |- ( F Fn A -> ( F " A ) = ran F )

Syntax hints:   -> wi 4   = wceq 1332   ran crn 4548   |` cres 4549   "cima 4550   Fn wfn 5126

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-xp 4553  df-rel 4554  df-cnv 4555  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-fun 5133  df-fn 5134

This theorem is referenced by:  f1finf1o  6843  tgrest  12377