Metamath Proof Explorer


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 : 𝐴1-1-onto𝐵𝜑 ) }
Assertion f1oabexg ( ( 𝐴𝐶𝐵𝐷 ) → 𝐹 ∈ V )

Proof

Step Hyp Ref Expression
1 f1oabexg.1 𝐹 = { 𝑓 ∣ ( 𝑓 : 𝐴1-1-onto𝐵𝜑 ) }
2 elex ( 𝐴𝐶𝐴 ∈ V )
3 elex ( 𝐵𝐷𝐵 ∈ V )
4 f1of ( 𝑓 : 𝐴1-1-onto𝐵𝑓 : 𝐴𝐵 )
5 4 ad2antrl ( ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) ∧ ( 𝑓 : 𝐴1-1-onto𝐵𝜑 ) ) → 𝑓 : 𝐴𝐵 )
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 : 𝐴1-1-onto𝐵𝜑 ) } ∈ V )
9 1 8 eqeltrid ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → 𝐹 ∈ V )
10 2 3 9 syl2an ( ( 𝐴𝐶𝐵𝐷 ) → 𝐹 ∈ V )