Theorem dffo2 5274

Description: Alternate definition of an onto function. (Contributed by set.mm contributors, 22-Mar-2006.)

Assertion

Ref	Expression
dffo2	⊢ (F:A–onto→B ↔ (F:A–→B ∧ ran F = B))

Proof of Theorem dffo2

Step	Hyp	Ref	Expression
1		fof 5270	. . 3 ⊢ (F:A–onto→B → F:A–→B)
2		forn 5273	. . 3 ⊢ (F:A–onto→B → ran F = B)
3	1, 2	jca 518	. 2 ⊢ (F:A–onto→B → (F:A–→B ∧ ran F = B))
4		ffn 5224	. . 3 ⊢ (F:A–→B → F Fn A)
5		df-fo 4794	. . . 4 ⊢ (F:A–onto→B ↔ (F Fn A ∧ ran F = B))
6	5	biimpri 197	. . 3 ⊢ ((F Fn A ∧ ran F = B) → F:A–onto→B)
7	4, 6	sylan 457	. 2 ⊢ ((F:A–→B ∧ ran F = B) → F:A–onto→B)
8	3, 7	impbii 180	1 ⊢ (F:A–onto→B ↔ (F:A–→B ∧ ran F = B))

Syntax hints:   ↔ wb 176   ∧ wa 358   = wceq 1642  ran crn 4774   Fn wfn 4777  –→wf 4778  –onto→wfo 4780

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-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-fo 4794

This theorem is referenced by:  foco  5280  foconst  5281  dff1o5  5296  dffo3  5423