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Metamath Proof Explorer

Theorem dff14a

Description: A one-to-one function in terms of different function values for different arguments. (Contributed by Alexander van der Vekens, 26-Jan-2018)

Ref Expression
Assertion dff14a ( 𝐹 : 𝐴 –1-1→ 𝐵 ↔ ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 ≠ 𝑦 → ( 𝐹 ‘ 𝑥 ) ≠ ( 𝐹 ‘ 𝑦 ) ) ) )

Proof

Step Hyp Ref Expression
1 dff13 ⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 ↔ ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) )
2 con34b ⊢ ( ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) → 𝑥 = 𝑦 ) ↔ ( ¬ 𝑥 = 𝑦 → ¬ ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ) )
3 df-ne ⊢ ( 𝑥 ≠ 𝑦 ↔ ¬ 𝑥 = 𝑦 )
4 3 bicomi ⊢ ( ¬ 𝑥 = 𝑦 ↔ 𝑥 ≠ 𝑦 )
5 df-ne ⊢ ( ( 𝐹 ‘ 𝑥 ) ≠ ( 𝐹 ‘ 𝑦 ) ↔ ¬ ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) )
6 5 bicomi ⊢ ( ¬ ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ↔ ( 𝐹 ‘ 𝑥 ) ≠ ( 𝐹 ‘ 𝑦 ) )
7 4 6 imbi12i ⊢ ( ( ¬ 𝑥 = 𝑦 → ¬ ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ) ↔ ( 𝑥 ≠ 𝑦 → ( 𝐹 ‘ 𝑥 ) ≠ ( 𝐹 ‘ 𝑦 ) ) )
8 2 7 bitri ⊢ ( ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) → 𝑥 = 𝑦 ) ↔ ( 𝑥 ≠ 𝑦 → ( 𝐹 ‘ 𝑥 ) ≠ ( 𝐹 ‘ 𝑦 ) ) )
9 8 2ralbii ⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) → 𝑥 = 𝑦 ) ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 ≠ 𝑦 → ( 𝐹 ‘ 𝑥 ) ≠ ( 𝐹 ‘ 𝑦 ) ) )
10 9 anbi2i ⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) ↔ ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 ≠ 𝑦 → ( 𝐹 ‘ 𝑥 ) ≠ ( 𝐹 ‘ 𝑦 ) ) ) )
11 1 10 bitri ⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 ↔ ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 ≠ 𝑦 → ( 𝐹 ‘ 𝑥 ) ≠ ( 𝐹 ‘ 𝑦 ) ) ) )