Theorem fnima 5451

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 4738	. 2 |- ( F " A ) = ran ( F |` A )
2		fnresdm 5441	. . 3 |- ( F Fn A -> ( F |` A ) = F )
3	2	rneqd 4961	. 2 |- ( F Fn A -> ran ( F |` A ) = ran F )
4	1, 3	eqtrid 2276	1 |- ( F Fn A -> ( F " A ) = ran F )

Syntax hints:   -> wi 4   = wceq 1397   ran crn 4726   |` cres 4727   "cima 4728   Fn wfn 5321

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-rel 4732  df-cnv 4733  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-fun 5328  df-fn 5329

This theorem is referenced by:  f1finf1o  7145  tgrest  14892