Theorem f1dmex

Description: If the codomain of a one-to-one function exists, so does its domain. This theorem is equivalent to the Axiom of Replacement ax-rep . (Contributed by NM, 4-Sep-2004)

	Ref	Expression
Assertion	f1dmex	⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ 𝐵 ∈ 𝐶 ) → 𝐴 ∈ V )

Proof

Step	Hyp	Ref	Expression
1		f1f	⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 → 𝐹 : 𝐴 ⟶ 𝐵 )
2	1	frnd	⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 → ran 𝐹 ⊆ 𝐵 )
3		ssexg	⊢ ( ( ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶 ) → ran 𝐹 ∈ V )
4	2 3	sylan	⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ 𝐵 ∈ 𝐶 ) → ran 𝐹 ∈ V )
5	4	ex	⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 → ( 𝐵 ∈ 𝐶 → ran 𝐹 ∈ V ) )
6		f1cnv	⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 → ◡ 𝐹 : ran 𝐹 –1-1-onto→ 𝐴 )
7		f1ofo	⊢ ( ◡ 𝐹 : ran 𝐹 –1-1-onto→ 𝐴 → ◡ 𝐹 : ran 𝐹 –onto→ 𝐴 )
8	6 7	syl	⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 → ◡ 𝐹 : ran 𝐹 –onto→ 𝐴 )
9		fornex	⊢ ( ran 𝐹 ∈ V → ( ◡ 𝐹 : ran 𝐹 –onto→ 𝐴 → 𝐴 ∈ V ) )
10	5 8 9	syl6ci	⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 → ( 𝐵 ∈ 𝐶 → 𝐴 ∈ V ) )
11	10	imp	⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ 𝐵 ∈ 𝐶 ) → 𝐴 ∈ V )