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Metamath Proof Explorer

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 focdmex ⊢ ( 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 )