Theorem f1oabexg

Description: The class of all 1-1-onto functions mapping one set to another is a set. (Contributed by Paul Chapman, 25-Feb-2008) (Proof shortened by AV, 9-Jun-2025)

	Ref	Expression
Hypothesis	f1oabexg.1	⊢ 𝐹 = { 𝑓 ∣ ( 𝑓 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝜑 ) }
Assertion	f1oabexg	⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ) → 𝐹 ∈ V )

Proof

Step	Hyp	Ref	Expression
1		f1oabexg.1	⊢ 𝐹 = { 𝑓 ∣ ( 𝑓 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝜑 ) }
2		elex	⊢ ( 𝐴 ∈ 𝐶 → 𝐴 ∈ V )
3		elex	⊢ ( 𝐵 ∈ 𝐷 → 𝐵 ∈ V )
4		f1of	⊢ ( 𝑓 : 𝐴 –1-1-onto→ 𝐵 → 𝑓 : 𝐴 ⟶ 𝐵 )
5	4	ad2antrl	⊢ ( ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) ∧ ( 𝑓 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝜑 ) ) → 𝑓 : 𝐴 ⟶ 𝐵 )
6		simpl	⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → 𝐴 ∈ V )
7		simpr	⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → 𝐵 ∈ V )
8	5 6 7	fabexd	⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → { 𝑓 ∣ ( 𝑓 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝜑 ) } ∈ V )
9	1 8	eqeltrid	⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → 𝐹 ∈ V )
10	2 3 9	syl2an	⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ) → 𝐹 ∈ V )