Theorem dff1o5 5295

Description: Alternate definition of one-to-one onto function. (The proof was shortened by Andrew Salmon, 22-Oct-2011.) (Contributed by set.mm contributors, 10-Dec-2003.) (Revised by set.mm contributors, 22-Oct-2011.)

Assertion

Ref	Expression
dff1o5	⊢ (F:A–1-1-onto→B ↔ (F:A–1-1→B ∧ ran F = B))

Proof of Theorem dff1o5

Step	Hyp	Ref	Expression
1		df-f1o 4794	. 2 ⊢ (F:A–1-1-onto→B ↔ (F:A–1-1→B ∧ F:A–onto→B))
2		f1f 5258	. . . . 5 ⊢ (F:A–1-1→B → F:A–→B)
3	2	biantrurd 494	. . . 4 ⊢ (F:A–1-1→B → (ran F = B ↔ (F:A–→B ∧ ran F = B)))
4		dffo2 5273	. . . 4 ⊢ (F:A–onto→B ↔ (F:A–→B ∧ ran F = B))
5	3, 4	syl6rbbr 255	. . 3 ⊢ (F:A–1-1→B → (F:A–onto→B ↔ ran F = B))
6	5	pm5.32i 618	. 2 ⊢ ((F:A–1-1→B ∧ F:A–onto→B) ↔ (F:A–1-1→B ∧ ran F = B))
7	1, 6	bitri 240	1 ⊢ (F:A–1-1-onto→B ↔ (F:A–1-1→B ∧ ran F = B))

Syntax hints:   ↔ wb 176   ∧ wa 358   = wceq 1642  ran crn 4773  –→wf 4777  –1-1→wf1 4778  –onto→wfo 4779  –1-1-onto→wf1o 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 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:  f1orescnv  5301