Theorem dff1o5 6824

Description: Alternate definition of one-to-one onto function. (Contributed by NM, 10-Dec-2003.) (Proof shortened by Andrew Salmon, 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 6535	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵))
2		dffo2 6791	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵))
3		f1f 6771	. . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
4	3	biantrurd 532	. . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → (ran 𝐹 = 𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵)))
5	2, 4	bitr4id 290	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → (𝐹:𝐴–onto→𝐵 ↔ ran 𝐹 = 𝐵))
6	5	pm5.32i 574	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵) ↔ (𝐹:𝐴–1-1→𝐵 ∧ ran 𝐹 = 𝐵))
7	1, 6	bitri 275	1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ ran 𝐹 = 𝐵))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1539  ran crn 5653  ⟶wf 6524  –1-1→wf1 6525  –onto→wfo 6526  –1-1-onto→wf1o 6527

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-cleq 2726  df-ss 3941  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535

This theorem is referenced by:  f1orescnv  6830  f1ounsn  7261  domdifsn  9063  sucdom2OLD  9091  sucdom2  9212  ackbij1  10244  ackbij2  10249  fin4en1  10316  om2uzf1oi  13961  s4f1o  14926  fvcosymgeq  19397  indlcim  21787  2lgslem1b  27341  ausgrusgrb  29078  usgrexmpledg  29175  wrdpmtrlast  33041  cdleme50f1o  40494  diaf1oN  41078  aks6d1c2  42072  pwssplit4  43045  cantnf2  43281  meadjiunlem  46430