Theorem f1ofo 5294

Description: A one-to-one onto function is an onto function. (Contributed by set.mm contributors, 28-Apr-2004.)

Assertion

Ref	Expression
f1ofo	⊢ (F:A–1-1-onto→B → F:A–onto→B)

Proof of Theorem f1ofo

Step	Hyp	Ref	Expression
1		dff1o3 5293	. 2 ⊢ (F:A–1-1-onto→B ↔ (F:A–onto→B ∧ Fun ◡F))
2	1	simplbi 446	1 ⊢ (F:A–1-1-onto→B → F:A–onto→B)

Syntax hints:   → wi 4  ^◡ccnv 4772  Fun wfun 4776  –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:  f1imacnv  5303  resin  5308  f1ococnv2  5310  fo00  5319  isoini  5498  bren  6031  enpw1  6063  enmap1lem5  6074  nenpw1pwlem2  6086  ncdisjun  6137  1cnc  6140  sbthlem3  6206  lenc  6224