Theorem f1domg

Description: The domain of a one-to-one function is dominated by its codomain. (Contributed by NM, 4-Sep-2004)

	Ref	Expression
Assertion	f1domg	⊢ ( 𝐵 ∈ 𝐶 → ( 𝐹 : 𝐴 –1-1→ 𝐵 → 𝐴 ≼ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		f1f	⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 → 𝐹 : 𝐴 ⟶ 𝐵 )
2		f1dmex	⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ 𝐵 ∈ 𝐶 ) → 𝐴 ∈ V )
3		fex	⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐴 ∈ V ) → 𝐹 ∈ V )
4	1 2 3	syl2an2r	⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ 𝐵 ∈ 𝐶 ) → 𝐹 ∈ V )
5	4	expcom	⊢ ( 𝐵 ∈ 𝐶 → ( 𝐹 : 𝐴 –1-1→ 𝐵 → 𝐹 ∈ V ) )
6		f1eq1	⊢ ( 𝑓 = 𝐹 → ( 𝑓 : 𝐴 –1-1→ 𝐵 ↔ 𝐹 : 𝐴 –1-1→ 𝐵 ) )
7	6	spcegv	⊢ ( 𝐹 ∈ V → ( 𝐹 : 𝐴 –1-1→ 𝐵 → ∃ 𝑓 𝑓 : 𝐴 –1-1→ 𝐵 ) )
8	5 7	syli	⊢ ( 𝐵 ∈ 𝐶 → ( 𝐹 : 𝐴 –1-1→ 𝐵 → ∃ 𝑓 𝑓 : 𝐴 –1-1→ 𝐵 ) )
9		brdomg	⊢ ( 𝐵 ∈ 𝐶 → ( 𝐴 ≼ 𝐵 ↔ ∃ 𝑓 𝑓 : 𝐴 –1-1→ 𝐵 ) )
10	8 9	sylibrd	⊢ ( 𝐵 ∈ 𝐶 → ( 𝐹 : 𝐴 –1-1→ 𝐵 → 𝐴 ≼ 𝐵 ) )