Theorem f1imaeq

Description: Taking images under a one-to-one function preserves equality. (Contributed by Stefan O'Rear, 30-Oct-2014)

	Ref	Expression
Assertion	f1imaeq	⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ ( 𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴 ) ) → ( ( 𝐹 “ 𝐶 ) = ( 𝐹 “ 𝐷 ) ↔ 𝐶 = 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		f1imass	⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ ( 𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴 ) ) → ( ( 𝐹 “ 𝐶 ) ⊆ ( 𝐹 “ 𝐷 ) ↔ 𝐶 ⊆ 𝐷 ) )
2		f1imass	⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ ( 𝐷 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴 ) ) → ( ( 𝐹 “ 𝐷 ) ⊆ ( 𝐹 “ 𝐶 ) ↔ 𝐷 ⊆ 𝐶 ) )
3	2	ancom2s	⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ ( 𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴 ) ) → ( ( 𝐹 “ 𝐷 ) ⊆ ( 𝐹 “ 𝐶 ) ↔ 𝐷 ⊆ 𝐶 ) )
4	1 3	anbi12d	⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ ( 𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴 ) ) → ( ( ( 𝐹 “ 𝐶 ) ⊆ ( 𝐹 “ 𝐷 ) ∧ ( 𝐹 “ 𝐷 ) ⊆ ( 𝐹 “ 𝐶 ) ) ↔ ( 𝐶 ⊆ 𝐷 ∧ 𝐷 ⊆ 𝐶 ) ) )
5		eqss	⊢ ( ( 𝐹 “ 𝐶 ) = ( 𝐹 “ 𝐷 ) ↔ ( ( 𝐹 “ 𝐶 ) ⊆ ( 𝐹 “ 𝐷 ) ∧ ( 𝐹 “ 𝐷 ) ⊆ ( 𝐹 “ 𝐶 ) ) )
6		eqss	⊢ ( 𝐶 = 𝐷 ↔ ( 𝐶 ⊆ 𝐷 ∧ 𝐷 ⊆ 𝐶 ) )
7	4 5 6	3bitr4g	⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ ( 𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴 ) ) → ( ( 𝐹 “ 𝐶 ) = ( 𝐹 “ 𝐷 ) ↔ 𝐶 = 𝐷 ) )