Theorem f1orn 5296

Description: A one-to-one function maps onto its range. (Contributed by set.mm contributors, 13-Aug-2004.)

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 936	. 2 |- ((F Fn A /\ Fun `'F /\ ran F = ran F) <-> ((F Fn A /\ Fun `'F) /\ ran F = ran F))
2		dff1o2 5291	. 2 |- (F:A-1-1-onto->ran F <-> (F Fn A /\ Fun `'F /\ ran F = ran F))
3		eqid 2353	. . 3 |- ran F = ran F
4	3	biantru 491	. 2 |- ((F Fn A /\ Fun `'F) <-> ((F Fn A /\ Fun `'F) /\ ran F = ran F))
5	1, 2, 4	3bitr4i 268	1 |- (F:A-1-1-onto->ran F <-> (F Fn A /\ Fun `'F))

Syntax hints:   <-> wb 176   /\ wa 358   /\ w3a 934   = wceq 1642  `'ccnv 4771  ran crn 4773  Fun wfun 4775   Fn wfn 4776  -1-1-onto->wf1o 4780

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794

This theorem is referenced by:  f1f1orn  5297