Theorem f1veqaeq

Description: If the values of a one-to-one function for two arguments are equal, the arguments themselves must be equal. (Contributed by Alexander van der Vekens, 12-Nov-2017)

	Ref	Expression
Assertion	f1veqaeq	⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ ( 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴 ) ) → ( ( 𝐹 ‘ 𝐶 ) = ( 𝐹 ‘ 𝐷 ) → 𝐶 = 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		dff13	⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 ↔ ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ∀ 𝑐 ∈ 𝐴 ∀ 𝑑 ∈ 𝐴 ( ( 𝐹 ‘ 𝑐 ) = ( 𝐹 ‘ 𝑑 ) → 𝑐 = 𝑑 ) ) )
2		fveqeq2	⊢ ( 𝑐 = 𝐶 → ( ( 𝐹 ‘ 𝑐 ) = ( 𝐹 ‘ 𝑑 ) ↔ ( 𝐹 ‘ 𝐶 ) = ( 𝐹 ‘ 𝑑 ) ) )
3		eqeq1	⊢ ( 𝑐 = 𝐶 → ( 𝑐 = 𝑑 ↔ 𝐶 = 𝑑 ) )
4	2 3	imbi12d	⊢ ( 𝑐 = 𝐶 → ( ( ( 𝐹 ‘ 𝑐 ) = ( 𝐹 ‘ 𝑑 ) → 𝑐 = 𝑑 ) ↔ ( ( 𝐹 ‘ 𝐶 ) = ( 𝐹 ‘ 𝑑 ) → 𝐶 = 𝑑 ) ) )
5		fveq2	⊢ ( 𝑑 = 𝐷 → ( 𝐹 ‘ 𝑑 ) = ( 𝐹 ‘ 𝐷 ) )
6	5	eqeq2d	⊢ ( 𝑑 = 𝐷 → ( ( 𝐹 ‘ 𝐶 ) = ( 𝐹 ‘ 𝑑 ) ↔ ( 𝐹 ‘ 𝐶 ) = ( 𝐹 ‘ 𝐷 ) ) )
7		eqeq2	⊢ ( 𝑑 = 𝐷 → ( 𝐶 = 𝑑 ↔ 𝐶 = 𝐷 ) )
8	6 7	imbi12d	⊢ ( 𝑑 = 𝐷 → ( ( ( 𝐹 ‘ 𝐶 ) = ( 𝐹 ‘ 𝑑 ) → 𝐶 = 𝑑 ) ↔ ( ( 𝐹 ‘ 𝐶 ) = ( 𝐹 ‘ 𝐷 ) → 𝐶 = 𝐷 ) ) )
9	4 8	rspc2v	⊢ ( ( 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴 ) → ( ∀ 𝑐 ∈ 𝐴 ∀ 𝑑 ∈ 𝐴 ( ( 𝐹 ‘ 𝑐 ) = ( 𝐹 ‘ 𝑑 ) → 𝑐 = 𝑑 ) → ( ( 𝐹 ‘ 𝐶 ) = ( 𝐹 ‘ 𝐷 ) → 𝐶 = 𝐷 ) ) )
10	9	com12	⊢ ( ∀ 𝑐 ∈ 𝐴 ∀ 𝑑 ∈ 𝐴 ( ( 𝐹 ‘ 𝑐 ) = ( 𝐹 ‘ 𝑑 ) → 𝑐 = 𝑑 ) → ( ( 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴 ) → ( ( 𝐹 ‘ 𝐶 ) = ( 𝐹 ‘ 𝐷 ) → 𝐶 = 𝐷 ) ) )
11	1 10	simplbiim	⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 → ( ( 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴 ) → ( ( 𝐹 ‘ 𝐶 ) = ( 𝐹 ‘ 𝐷 ) → 𝐶 = 𝐷 ) ) )
12	11	imp	⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ ( 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴 ) ) → ( ( 𝐹 ‘ 𝐶 ) = ( 𝐹 ‘ 𝐷 ) → 𝐶 = 𝐷 ) )