Theorem wunress

Description: Closure of structure restriction in a weak universe. (Contributed by Mario Carneiro, 12-Jan-2017) (Proof shortened by AV, 28-Oct-2024)

	Ref	Expression
Hypotheses	wunress.1	⊢ ( 𝜑 → 𝑈 ∈ WUni )
	wunress.2	⊢ ( 𝜑 → ω ∈ 𝑈 )
	wunress.3	⊢ ( 𝜑 → 𝑊 ∈ 𝑈 )
Assertion	wunress	⊢ ( 𝜑 → ( 𝑊 ↾s 𝐴 ) ∈ 𝑈 )

Proof

Step	Hyp	Ref	Expression
1		wunress.1	⊢ ( 𝜑 → 𝑈 ∈ WUni )
2		wunress.2	⊢ ( 𝜑 → ω ∈ 𝑈 )
3		wunress.3	⊢ ( 𝜑 → 𝑊 ∈ 𝑈 )
4		eqid	⊢ ( 𝑊 ↾s 𝐴 ) = ( 𝑊 ↾s 𝐴 )
5		eqid	⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 )
6	4 5	ressval	⊢ ( ( 𝑊 ∈ 𝑈 ∧ 𝐴 ∈ V ) → ( 𝑊 ↾s 𝐴 ) = if ( ( Base ‘ 𝑊 ) ⊆ 𝐴 , 𝑊 , ( 𝑊 sSet ⟨ ( Base ‘ ndx ) , ( 𝐴 ∩ ( Base ‘ 𝑊 ) ) ⟩ ) ) )
7	3 6	sylan	⊢ ( ( 𝜑 ∧ 𝐴 ∈ V ) → ( 𝑊 ↾s 𝐴 ) = if ( ( Base ‘ 𝑊 ) ⊆ 𝐴 , 𝑊 , ( 𝑊 sSet ⟨ ( Base ‘ ndx ) , ( 𝐴 ∩ ( Base ‘ 𝑊 ) ) ⟩ ) ) )
8	1 2	basndxelwund	⊢ ( 𝜑 → ( Base ‘ ndx ) ∈ 𝑈 )
9		incom	⊢ ( 𝐴 ∩ ( Base ‘ 𝑊 ) ) = ( ( Base ‘ 𝑊 ) ∩ 𝐴 )
10		baseid	⊢ Base = Slot ( Base ‘ ndx )
11	10 1 3	wunstr	⊢ ( 𝜑 → ( Base ‘ 𝑊 ) ∈ 𝑈 )
12	1 11	wunin	⊢ ( 𝜑 → ( ( Base ‘ 𝑊 ) ∩ 𝐴 ) ∈ 𝑈 )
13	9 12	eqeltrid	⊢ ( 𝜑 → ( 𝐴 ∩ ( Base ‘ 𝑊 ) ) ∈ 𝑈 )
14	1 8 13	wunop	⊢ ( 𝜑 → ⟨ ( Base ‘ ndx ) , ( 𝐴 ∩ ( Base ‘ 𝑊 ) ) ⟩ ∈ 𝑈 )
15	1 3 14	wunsets	⊢ ( 𝜑 → ( 𝑊 sSet ⟨ ( Base ‘ ndx ) , ( 𝐴 ∩ ( Base ‘ 𝑊 ) ) ⟩ ) ∈ 𝑈 )
16	3 15	ifcld	⊢ ( 𝜑 → if ( ( Base ‘ 𝑊 ) ⊆ 𝐴 , 𝑊 , ( 𝑊 sSet ⟨ ( Base ‘ ndx ) , ( 𝐴 ∩ ( Base ‘ 𝑊 ) ) ⟩ ) ) ∈ 𝑈 )
17	16	adantr	⊢ ( ( 𝜑 ∧ 𝐴 ∈ V ) → if ( ( Base ‘ 𝑊 ) ⊆ 𝐴 , 𝑊 , ( 𝑊 sSet ⟨ ( Base ‘ ndx ) , ( 𝐴 ∩ ( Base ‘ 𝑊 ) ) ⟩ ) ) ∈ 𝑈 )
18	7 17	eqeltrd	⊢ ( ( 𝜑 ∧ 𝐴 ∈ V ) → ( 𝑊 ↾s 𝐴 ) ∈ 𝑈 )
19	18	ex	⊢ ( 𝜑 → ( 𝐴 ∈ V → ( 𝑊 ↾s 𝐴 ) ∈ 𝑈 ) )
20	1	wun0	⊢ ( 𝜑 → ∅ ∈ 𝑈 )
21		reldmress	⊢ Rel dom ↾s
22	21	ovprc2	⊢ ( ¬ 𝐴 ∈ V → ( 𝑊 ↾s 𝐴 ) = ∅ )
23	22	eleq1d	⊢ ( ¬ 𝐴 ∈ V → ( ( 𝑊 ↾s 𝐴 ) ∈ 𝑈 ↔ ∅ ∈ 𝑈 ) )
24	20 23	syl5ibrcom	⊢ ( 𝜑 → ( ¬ 𝐴 ∈ V → ( 𝑊 ↾s 𝐴 ) ∈ 𝑈 ) )
25	19 24	pm2.61d	⊢ ( 𝜑 → ( 𝑊 ↾s 𝐴 ) ∈ 𝑈 )