Theorem dff1o3 5486

Description: Alternate definition of one-to-one onto function. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 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		3anan32 991	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = 𝐵) ↔ ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) ∧ Fun ◡𝐹))
2		dff1o2 5485	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = 𝐵))
3		df-fo 5241	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
4	3	anbi1i 458	. 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ Fun ◡𝐹) ↔ ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) ∧ Fun ◡𝐹))
5	1, 2, 4	3bitr4i 212	1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–onto→𝐵 ∧ Fun ◡𝐹))

Syntax hints:   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1364  ^◡ccnv 4643  ran crn 4645  Fun wfun 5229   Fn wfn 5230  –onto→wfo 5233  –1-1-onto→wf1o 5234

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-3an 982  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-in 3150  df-ss 3157  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242

This theorem is referenced by:  f1ofo  5487  resdif  5502  f11o  5513  f1opw  6102  1stconst  6247  2ndconst  6248  f1o2ndf1  6254  ssdomg  6805  phplem4  6884  phplem4on  6896