Theorem f1orn 3683

Description: A one-to-one function maps onto its range.

Assertion

Ref	Expression
f1orn	|- (F:A-1-1-onto->ran F <-> (F Fn A /\ Fun `'F))

Proof of Theorem f1orn

Step	Hyp	Ref	Expression
1		df-3an 775	. 2 |- ((F Fn A /\ Fun `'F /\ ran F = ran F) <-> ((F Fn A /\ Fun `'F) /\ ran F = ran F))
2		f1o2 3678	. 2 |- (F:A-1-1-onto->ran F <-> (F Fn A /\ Fun `'F /\ ran F = ran F))
3		eqid 1468	. . 3 |- ran F = ran F
4	3	biantru 722	. 2 |- ((F Fn A /\ Fun `'F) <-> ((F Fn A /\ Fun `'F) /\ ran F = ran F))
5	1, 2, 4	3bitr4 183	1 |- (F:A-1-1-onto->ran F <-> (F Fn A /\ Fun `'F))

Syntax hints:   <-> wb 146   /\ wa 223   /\ w3a 773   = wceq 953  `'ccnv 3159  ran crn 3161  Fun wfun 3166   Fn wfn 3167  -1-1-onto->wf1o 3171

This theorem is referenced by:  f1f1orn 3684

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-in 2041  df-ss 2043  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187

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