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 ) ) )