Theorem f1orn 3704

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 779	. 2 |- ((F Fn A /\ Fun `'F /\ ran F = ran F) <-> ((F Fn A /\ Fun `'F) /\ ran F = ran F))
2		f1o2 3699	. 2 |- (F:A-1-1-onto->ran F <-> (F Fn A /\ Fun `'F /\ ran F = ran F))
3		eqid 1478	. . 3 |- ran F = ran F
4	3	biantru 726	. 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 777   = wceq 958  `'ccnv 3175  ran crn 3177  Fun wfun 3182   Fn wfn 3183  -1-1-onto->wf1o 3187

This theorem is referenced by:  f1f1orn 3705

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-in 2054  df-ss 2056  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203

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