Theorem f1imapss

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

	Ref	Expression
Assertion	f1imapss	⊢ ( ( 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		f1imaeq	⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ ( 𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴 ) ) → ( ( 𝐹 “ 𝐶 ) = ( 𝐹 “ 𝐷 ) ↔ 𝐶 = 𝐷 ) )
3	2	notbid	⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ ( 𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴 ) ) → ( ¬ ( 𝐹 “ 𝐶 ) = ( 𝐹 “ 𝐷 ) ↔ ¬ 𝐶 = 𝐷 ) )
4	1 3	anbi12d	⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ ( 𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴 ) ) → ( ( ( 𝐹 “ 𝐶 ) ⊆ ( 𝐹 “ 𝐷 ) ∧ ¬ ( 𝐹 “ 𝐶 ) = ( 𝐹 “ 𝐷 ) ) ↔ ( 𝐶 ⊆ 𝐷 ∧ ¬ 𝐶 = 𝐷 ) ) )
5		dfpss2	⊢ ( ( 𝐹 “ 𝐶 ) ⊊ ( 𝐹 “ 𝐷 ) ↔ ( ( 𝐹 “ 𝐶 ) ⊆ ( 𝐹 “ 𝐷 ) ∧ ¬ ( 𝐹 “ 𝐶 ) = ( 𝐹 “ 𝐷 ) ) )
6		dfpss2	⊢ ( 𝐶 ⊊ 𝐷 ↔ ( 𝐶 ⊆ 𝐷 ∧ ¬ 𝐶 = 𝐷 ) )
7	4 5 6	3bitr4g	⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ ( 𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴 ) ) → ( ( 𝐹 “ 𝐶 ) ⊊ ( 𝐹 “ 𝐷 ) ↔ 𝐶 ⊊ 𝐷 ) )