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Metamath Proof Explorer

Theorem f1oen3g

Description: The domain and range of a one-to-one, onto set function are equinumerous. This variation of f1oeng does not require the Axiom of Replacement nor the Axiom of Power Sets. (Contributed by NM, 13-Jan-2007) (Revised by Mario Carneiro, 10-Sep-2015)

Ref Expression
Assertion f1oen3g ( ( 𝐹 ∈ 𝑉 ∧ 𝐹 : 𝐴 –1-1-onto→ 𝐵 ) → 𝐴 ≈ 𝐵 )

Proof

Step Hyp Ref Expression
1 f1oeq1 ⊢ ( 𝑓 = 𝐹 → ( 𝑓 : 𝐴 –1-1-onto→ 𝐵 ↔ 𝐹 : 𝐴 –1-1-onto→ 𝐵 ) )
2 1 spcegv ⊢ ( 𝐹 ∈ 𝑉 → ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 → ∃ 𝑓 𝑓 : 𝐴 –1-1-onto→ 𝐵 ) )
3 2 imp ⊢ ( ( 𝐹 ∈ 𝑉 ∧ 𝐹 : 𝐴 –1-1-onto→ 𝐵 ) → ∃ 𝑓 𝑓 : 𝐴 –1-1-onto→ 𝐵 )
4 bren ⊢ ( 𝐴 ≈ 𝐵 ↔ ∃ 𝑓 𝑓 : 𝐴 –1-1-onto→ 𝐵 )
5 3 4 sylibr ⊢ ( ( 𝐹 ∈ 𝑉 ∧ 𝐹 : 𝐴 –1-1-onto→ 𝐵 ) → 𝐴 ≈ 𝐵 )