Theorem foima 5462

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 4998	. 2 |- ( F " dom F ) = ran F
2		fof 5457	. . . 4 |- ( F : A -onto-> B -> F : A --> B )
3		fdm 5390	. . . 4 |- ( F : A --> B -> dom F = A )
4	2, 3	syl 14	. . 3 |- ( F : A -onto-> B -> dom F = A )
5	4	imaeq2d 4988	. 2 |- ( F : A -onto-> B -> ( F " dom F ) = ( F " A ) )
6		forn 5460	. 2 |- ( F : A -onto-> B -> ran F = B )
7	1, 5, 6	3eqtr3a 2246	1 |- ( F : A -onto-> B -> ( F " A ) = B )

Syntax hints:   -> wi 4   = wceq 1364   dom cdm 4644   ran crn 4645   "cima 4647   -->wf 5231   -onto->wfo 5233

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-xp 4650  df-cnv 4652  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-fn 5238  df-f 5239  df-fo 5241

This theorem is referenced by:  foimacnv  5498  foima2  5773  fiintim  6957  fidcenumlemr  6984