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