Theorem f1oen2g

Description: The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 10-Sep-2015)

	Ref	Expression
Assertion	f1oen2g	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹 : 𝐴 –1-1-onto→ 𝐵 ) → 𝐴 ≈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		f1of	⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 → 𝐹 : 𝐴 ⟶ 𝐵 )
2		fex2	⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → 𝐹 ∈ V )
3	1 2	syl3an1	⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → 𝐹 ∈ V )
4	3	3coml	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹 : 𝐴 –1-1-onto→ 𝐵 ) → 𝐹 ∈ V )
5		simp3	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹 : 𝐴 –1-1-onto→ 𝐵 ) → 𝐹 : 𝐴 –1-1-onto→ 𝐵 )
6		f1oen3g	⊢ ( ( 𝐹 ∈ V ∧ 𝐹 : 𝐴 –1-1-onto→ 𝐵 ) → 𝐴 ≈ 𝐵 )
7	4 5 6	syl2anc	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹 : 𝐴 –1-1-onto→ 𝐵 ) → 𝐴 ≈ 𝐵 )