Theorem fnima 5414

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 4706	. 2 |- ( F " A ) = ran ( F |` A )
2		fnresdm 5404	. . 3 |- ( F Fn A -> ( F |` A ) = F )
3	2	rneqd 4926	. 2 |- ( F Fn A -> ran ( F |` A ) = ran F )
4	1, 3	eqtrid 2252	1 |- ( F Fn A -> ( F " A ) = ran F )

Syntax hints:   -> wi 4   = wceq 1373   ran crn 4694   |` cres 4695   "cima 4696   Fn wfn 5285

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-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-xp 4699  df-rel 4700  df-cnv 4701  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-fun 5292  df-fn 5293

This theorem is referenced by:  f1finf1o  7075  tgrest  14756