Theorem fnima 5285

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 4596	. 2 |- ( F " A ) = ran ( F |` A )
2		fnresdm 5276	. . 3 |- ( F Fn A -> ( F |` A ) = F )
3	2	rneqd 4812	. 2 |- ( F Fn A -> ran ( F |` A ) = ran F )
4	1, 3	syl5eq 2202	1 |- ( F Fn A -> ( F " A ) = ran F )

Syntax hints:   -> wi 4   = wceq 1335   ran crn 4584   |` cres 4585   "cima 4586   Fn wfn 5162

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4134  ax-pr 4168

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-xp 4589  df-rel 4590  df-cnv 4591  df-dm 4593  df-rn 4594  df-res 4595  df-ima 4596  df-fun 5169  df-fn 5170

This theorem is referenced by:  f1finf1o  6884  tgrest  12529