Theorem oprabexd

Description: Existence of an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009) (Revised by AV, 9-Aug-2024)

	Ref	Expression
Hypotheses	oprabexd.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
	oprabexd.2	⊢ ( 𝜑 → 𝐵 ∈ 𝑊 )
	oprabexd.3	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) → ∃* 𝑧 𝜓 )
	oprabexd.4	⊢ ( 𝜑 → 𝐹 = { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝜓 ) } )
Assertion	oprabexd	⊢ ( 𝜑 → 𝐹 ∈ V )

Proof

Step	Hyp	Ref	Expression
1		oprabexd.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
2		oprabexd.2	⊢ ( 𝜑 → 𝐵 ∈ 𝑊 )
3		oprabexd.3	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) → ∃* 𝑧 𝜓 )
4		oprabexd.4	⊢ ( 𝜑 → 𝐹 = { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝜓 ) } )
5	3	ex	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) → ∃* 𝑧 𝜓 ) )
6		moanimv	⊢ ( ∃* 𝑧 ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝜓 ) ↔ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) → ∃* 𝑧 𝜓 ) )
7	5 6	sylibr	⊢ ( 𝜑 → ∃* 𝑧 ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝜓 ) )
8	7	alrimivv	⊢ ( 𝜑 → ∀ 𝑥 ∀ 𝑦 ∃* 𝑧 ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝜓 ) )
9		funoprabg	⊢ ( ∀ 𝑥 ∀ 𝑦 ∃* 𝑧 ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝜓 ) → Fun { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝜓 ) } )
10	8 9	syl	⊢ ( 𝜑 → Fun { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝜓 ) } )
11		dmoprabss	⊢ dom { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝜓 ) } ⊆ ( 𝐴 × 𝐵 )
12	1 2	xpexd	⊢ ( 𝜑 → ( 𝐴 × 𝐵 ) ∈ V )
13		ssexg	⊢ ( ( dom { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝜓 ) } ⊆ ( 𝐴 × 𝐵 ) ∧ ( 𝐴 × 𝐵 ) ∈ V ) → dom { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝜓 ) } ∈ V )
14	11 12 13	sylancr	⊢ ( 𝜑 → dom { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝜓 ) } ∈ V )
15		funex	⊢ ( ( Fun { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝜓 ) } ∧ dom { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝜓 ) } ∈ V ) → { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝜓 ) } ∈ V )
16	10 14 15	syl2anc	⊢ ( 𝜑 → { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝜓 ) } ∈ V )
17	4 16	eqeltrd	⊢ ( 𝜑 → 𝐹 ∈ V )