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Metamath Proof Explorer

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 ) ) )