Theorem dff1o3 5293

Description: Alternate definition of one-to-one onto function. (The proof was shortened by Andrew Salmon, 22-Oct-2011.) (Contributed by set.mm contributors, 25-Mar-1998.) (Revised by set.mm contributors, 22-Oct-2011.)

Assertion

Ref	Expression
dff1o3	|- (F:A-1-1-onto->B <-> (F:A-onto->B /\ Fun `'F))

Proof of Theorem dff1o3

Step	Hyp	Ref	Expression
1		df-3an 936	. . 3 |- ((F Fn A /\ Fun `'F /\ ran F = B) <-> ((F Fn A /\ Fun `'F) /\ ran F = B))
2		an32 773	. . 3 |- (((F Fn A /\ Fun `'F) /\ ran F = B) <-> ((F Fn A /\ ran F = B) /\ Fun `'F))
3	1, 2	bitri 240	. 2 |- ((F Fn A /\ Fun `'F /\ ran F = B) <-> ((F Fn A /\ ran F = B) /\ Fun `'F))
4		dff1o2 5292	. 2 |- (F:A-1-1-onto->B <-> (F Fn A /\ Fun `'F /\ ran F = B))
5		df-fo 4794	. . 3 |- (F:A-onto->B <-> (F Fn A /\ ran F = B))
6	5	anbi1i 676	. 2 |- ((F:A-onto->B /\ Fun `'F) <-> ((F Fn A /\ ran F = B) /\ Fun `'F))
7	3, 4, 6	3bitr4i 268	1 |- (F:A-1-1-onto->B <-> (F:A-onto->B /\ Fun `'F))

Syntax hints:   <-> wb 176   /\ wa 358   /\ w3a 934   = wceq 1642  `'ccnv 4772  ran crn 4774  Fun wfun 4776   Fn wfn 4777  -onto->wfo 4780  -1-1-onto->wf1o 4781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795

This theorem is referenced by:  f1ofo  5294  f1ores  5301  resdif  5307  f11o  5316