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Metamath Proof Explorer

Theorem dff14b

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 dff14b ( 𝐹 : 𝐴 –1-1→ 𝐵 ↔ ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ ( 𝐴 ∖ { 𝑥 } ) ( 𝐹 ‘ 𝑥 ) ≠ ( 𝐹 ‘ 𝑦 ) ) )

Proof

Step Hyp Ref Expression
1 dff14a ⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 ↔ ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 ≠ 𝑦 → ( 𝐹 ‘ 𝑥 ) ≠ ( 𝐹 ‘ 𝑦 ) ) ) )
2 necom ⊢ ( 𝑥 ≠ 𝑦 ↔ 𝑦 ≠ 𝑥 )
3 2 imbi1i ⊢ ( ( 𝑥 ≠ 𝑦 → ( 𝐹 ‘ 𝑥 ) ≠ ( 𝐹 ‘ 𝑦 ) ) ↔ ( 𝑦 ≠ 𝑥 → ( 𝐹 ‘ 𝑥 ) ≠ ( 𝐹 ‘ 𝑦 ) ) )
4 3 ralbii ⊢ ( ∀ 𝑦 ∈ 𝐴 ( 𝑥 ≠ 𝑦 → ( 𝐹 ‘ 𝑥 ) ≠ ( 𝐹 ‘ 𝑦 ) ) ↔ ∀ 𝑦 ∈ 𝐴 ( 𝑦 ≠ 𝑥 → ( 𝐹 ‘ 𝑥 ) ≠ ( 𝐹 ‘ 𝑦 ) ) )
5 raldifsnb ⊢ ( ∀ 𝑦 ∈ 𝐴 ( 𝑦 ≠ 𝑥 → ( 𝐹 ‘ 𝑥 ) ≠ ( 𝐹 ‘ 𝑦 ) ) ↔ ∀ 𝑦 ∈ ( 𝐴 ∖ { 𝑥 } ) ( 𝐹 ‘ 𝑥 ) ≠ ( 𝐹 ‘ 𝑦 ) )
6 4 5 bitri ⊢ ( ∀ 𝑦 ∈ 𝐴 ( 𝑥 ≠ 𝑦 → ( 𝐹 ‘ 𝑥 ) ≠ ( 𝐹 ‘ 𝑦 ) ) ↔ ∀ 𝑦 ∈ ( 𝐴 ∖ { 𝑥 } ) ( 𝐹 ‘ 𝑥 ) ≠ ( 𝐹 ‘ 𝑦 ) )
7 6 ralbii ⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 ≠ 𝑦 → ( 𝐹 ‘ 𝑥 ) ≠ ( 𝐹 ‘ 𝑦 ) ) ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ ( 𝐴 ∖ { 𝑥 } ) ( 𝐹 ‘ 𝑥 ) ≠ ( 𝐹 ‘ 𝑦 ) )
8 7 anbi2i ⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 ≠ 𝑦 → ( 𝐹 ‘ 𝑥 ) ≠ ( 𝐹 ‘ 𝑦 ) ) ) ↔ ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ ( 𝐴 ∖ { 𝑥 } ) ( 𝐹 ‘ 𝑥 ) ≠ ( 𝐹 ‘ 𝑦 ) ) )
9 1 8 bitri ⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 ↔ ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ ( 𝐴 ∖ { 𝑥 } ) ( 𝐹 ‘ 𝑥 ) ≠ ( 𝐹 ‘ 𝑦 ) ) )