Theorem foima 5564

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 5086	. 2 |- ( F " dom F ) = ran F
2		fof 5559	. . . 4 |- ( F : A -onto-> B -> F : A --> B )
3		fdm 5488	. . . 4 |- ( F : A --> B -> dom F = A )
4	2, 3	syl 14	. . 3 |- ( F : A -onto-> B -> dom F = A )
5	4	imaeq2d 5076	. 2 |- ( F : A -onto-> B -> ( F " dom F ) = ( F " A ) )
6		forn 5562	. 2 |- ( F : A -onto-> B -> ran F = B )
7	1, 5, 6	3eqtr3a 2288	1 |- ( F : A -onto-> B -> ( F " A ) = B )

Syntax hints:   -> wi 4   = wceq 1397   dom cdm 4725   ran crn 4726   "cima 4728   -->wf 5322   -onto->wfo 5324

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-eu 2082  df-mo 2083  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-cnv 4733  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-fn 5329  df-f 5330  df-fo 5332

This theorem is referenced by:  foimacnv  5601  foima2  5891  fiintim  7122  fidcenumlemr  7153