Theorem foima 5601

Description: The image of the domain of an onto function. (Contributed by NM, 29-Nov-2002.)

Assertion

Ref	Expression
foima	|- ( F : A -onto-> B -> ( F " A ) = B )

Proof of Theorem foima

Step	Hyp	Ref	Expression
1		imadmrn 5117	. 2 |- ( F " dom F ) = ran F
2		fof 5596	. . . 4 |- ( F : A -onto-> B -> F : A --> B )
3		fdm 5520	. . . 4 |- ( F : A --> B -> dom F = A )
4	2, 3	syl 14	. . 3 |- ( F : A -onto-> B -> dom F = A )
5	4	imaeq2d 5107	. 2 |- ( F : A -onto-> B -> ( F " dom F ) = ( F " A ) )
6		forn 5599	. 2 |- ( F : A -onto-> B -> ran F = B )
7	1, 5, 6	3eqtr3a 2291	1 |- ( F : A -onto-> B -> ( F " A ) = B )

Syntax hints:   -> wi 4   = wceq 1398   dom cdm 4755   ran crn 4756   "cima 4758   -->wf 5354   -onto->wfo 5356

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4234  ax-pow 4293  ax-pr 4328

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-br 4116  df-opab 4178  df-xp 4761  df-cnv 4763  df-dm 4765  df-rn 4766  df-res 4767  df-ima 4768  df-fn 5361  df-f 5362  df-fo 5364

This theorem is referenced by:  foimacnv  5638  foima2  5931  fiintim  7205  fidcenumlemr  7239  eupthvdres  16601