Theorem dff15

Description: A one-to-one function in terms of different arguments never having the same function value. (Contributed by BTernaryTau, 24-Oct-2023)

	Ref	Expression
Assertion	dff15	⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 ↔ ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ¬ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ∧ 𝑥 ≠ 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		iman	⊢ ( ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) → 𝑥 = 𝑦 ) ↔ ¬ ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ∧ ¬ 𝑥 = 𝑦 ) )
3		df-ne	⊢ ( 𝑥 ≠ 𝑦 ↔ ¬ 𝑥 = 𝑦 )
4	3	anbi2i	⊢ ( ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ∧ 𝑥 ≠ 𝑦 ) ↔ ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ∧ ¬ 𝑥 = 𝑦 ) )
5	2 4	xchbinxr	⊢ ( ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) → 𝑥 = 𝑦 ) ↔ ¬ ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ∧ 𝑥 ≠ 𝑦 ) )
6	5	2ralbii	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) → 𝑥 = 𝑦 ) ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ∧ 𝑥 ≠ 𝑦 ) )
7		ralnex2	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ∧ 𝑥 ≠ 𝑦 ) ↔ ¬ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ∧ 𝑥 ≠ 𝑦 ) )
8	6 7	bitri	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) → 𝑥 = 𝑦 ) ↔ ¬ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ∧ 𝑥 ≠ 𝑦 ) )
9	8	anbi2i	⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) ↔ ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ¬ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ∧ 𝑥 ≠ 𝑦 ) ) )
10	1 9	bitri	⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 ↔ ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ¬ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ∧ 𝑥 ≠ 𝑦 ) ) )