Theorem dff1o6

Description: A one-to-one onto function in terms of function values. (Contributed by NM, 29-Mar-2008)

	Ref	Expression
Assertion	dff1o6	⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ↔ ( 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) )

Proof

Step	Hyp	Ref	Expression
1		df-f1o	⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ↔ ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ 𝐹 : 𝐴 –onto→ 𝐵 ) )
2		dff13	⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 ↔ ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) )
3		df-fo	⊢ ( 𝐹 : 𝐴 –onto→ 𝐵 ↔ ( 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵 ) )
4	2 3	anbi12i	⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ 𝐹 : 𝐴 –onto→ 𝐵 ) ↔ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) ∧ ( 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵 ) ) )
5		df-3an	⊢ ( ( 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) ↔ ( ( 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵 ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) )
6		eqimss	⊢ ( ran 𝐹 = 𝐵 → ran 𝐹 ⊆ 𝐵 )
7	6	anim2i	⊢ ( ( 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵 ) → ( 𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵 ) )
8		df-f	⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 ↔ ( 𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵 ) )
9	7 8	sylibr	⊢ ( ( 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵 ) → 𝐹 : 𝐴 ⟶ 𝐵 )
10	9	pm4.71ri	⊢ ( ( 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵 ) ↔ ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ( 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵 ) ) )
11	10	anbi1i	⊢ ( ( ( 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵 ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) ↔ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ( 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵 ) ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) )
12		an32	⊢ ( ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ( 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵 ) ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) ↔ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) ∧ ( 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵 ) ) )
13	5 11 12	3bitrri	⊢ ( ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) ∧ ( 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵 ) ) ↔ ( 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) )
14	1 4 13	3bitri	⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ↔ ( 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) )