Theorem wunress

Description: Closure of structure restriction in a weak universe. (Contributed by Mario Carneiro, 12-Jan-2017)

	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		df-base	⊢ Base = Slot 1
9	1 2	wunndx	⊢ ( 𝜑 → ndx ∈ 𝑈 )
10	8 1 9	wunstr	⊢ ( 𝜑 → ( Base ‘ ndx ) ∈ 𝑈 )
11		incom	⊢ ( 𝐴 ∩ ( Base ‘ 𝑊 ) ) = ( ( Base ‘ 𝑊 ) ∩ 𝐴 )
12	8 1 3	wunstr	⊢ ( 𝜑 → ( Base ‘ 𝑊 ) ∈ 𝑈 )
13	1 12	wunin	⊢ ( 𝜑 → ( ( Base ‘ 𝑊 ) ∩ 𝐴 ) ∈ 𝑈 )
14	11 13	eqeltrid	⊢ ( 𝜑 → ( 𝐴 ∩ ( Base ‘ 𝑊 ) ) ∈ 𝑈 )
15	1 10 14	wunop	⊢ ( 𝜑 → ⟨ ( Base ‘ ndx ) , ( 𝐴 ∩ ( Base ‘ 𝑊 ) ) ⟩ ∈ 𝑈 )
16	1 3 15	wunsets	⊢ ( 𝜑 → ( 𝑊 sSet ⟨ ( Base ‘ ndx ) , ( 𝐴 ∩ ( Base ‘ 𝑊 ) ) ⟩ ) ∈ 𝑈 )
17	3 16	ifcld	⊢ ( 𝜑 → if ( ( Base ‘ 𝑊 ) ⊆ 𝐴 , 𝑊 , ( 𝑊 sSet ⟨ ( Base ‘ ndx ) , ( 𝐴 ∩ ( Base ‘ 𝑊 ) ) ⟩ ) ) ∈ 𝑈 )
18	17	adantr	⊢ ( ( 𝜑 ∧ 𝐴 ∈ V ) → if ( ( Base ‘ 𝑊 ) ⊆ 𝐴 , 𝑊 , ( 𝑊 sSet ⟨ ( Base ‘ ndx ) , ( 𝐴 ∩ ( Base ‘ 𝑊 ) ) ⟩ ) ) ∈ 𝑈 )
19	7 18	eqeltrd	⊢ ( ( 𝜑 ∧ 𝐴 ∈ V ) → ( 𝑊 ↾s 𝐴 ) ∈ 𝑈 )
20	19	ex	⊢ ( 𝜑 → ( 𝐴 ∈ V → ( 𝑊 ↾s 𝐴 ) ∈ 𝑈 ) )
21	1	wun0	⊢ ( 𝜑 → ∅ ∈ 𝑈 )
22		reldmress	⊢ Rel dom ↾s
23	22	ovprc2	⊢ ( ¬ 𝐴 ∈ V → ( 𝑊 ↾s 𝐴 ) = ∅ )
24	23	eleq1d	⊢ ( ¬ 𝐴 ∈ V → ( ( 𝑊 ↾s 𝐴 ) ∈ 𝑈 ↔ ∅ ∈ 𝑈 ) )
25	21 24	syl5ibrcom	⊢ ( 𝜑 → ( ¬ 𝐴 ∈ V → ( 𝑊 ↾s 𝐴 ) ∈ 𝑈 ) )
26	20 25	pm2.61d	⊢ ( 𝜑 → ( 𝑊 ↾s 𝐴 ) ∈ 𝑈 )