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Theorem ressress 17200

Description: Restriction composition law. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Proof shortened by Mario Carneiro, 2-Dec-2014.)

**Assertion**
Ref	Expression
ressress	⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → ((𝑊 ↾_s 𝐴) ↾_s 𝐵) = (𝑊 ↾_s (𝐴 ∩ 𝐵)))

**Proof of Theorem ressress**
Step	Hyp	Ref	Expression
1		simplr 766	. . . . . . . . 9 ⊢ (((¬ (Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ¬ (Base‘𝑊) ⊆ 𝐴)
2		simpr1 1193	. . . . . . . . 9 ⊢ (((¬ (Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → 𝑊 ∈ V)
3		simpr2 1194	. . . . . . . . 9 ⊢ (((¬ (Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → 𝐴 ∈ 𝑋)
4		eqid 2731	. . . . . . . . . 10 ⊢ (𝑊 ↾_s 𝐴) = (𝑊 ↾_s 𝐴)
5		eqid 2731	. . . . . . . . . 10 ⊢ (Base‘𝑊) = (Base‘𝑊)
6	4, 5	ressval2 17185	. . . . . . . . 9 ⊢ ((¬ (Base‘𝑊) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑋) → (𝑊 ↾_s 𝐴) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩))
7	1, 2, 3, 6	syl3anc 1370	. . . . . . . 8 ⊢ (((¬ (Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑊 ↾_s 𝐴) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩))
8		inass 4219	. . . . . . . . . . 11 ⊢ ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊)) = (𝐴 ∩ (𝐵 ∩ (Base‘𝑊)))
9		in12 4220	. . . . . . . . . . 11 ⊢ (𝐴 ∩ (𝐵 ∩ (Base‘𝑊))) = (𝐵 ∩ (𝐴 ∩ (Base‘𝑊)))
10	8, 9	eqtri 2759	. . . . . . . . . 10 ⊢ ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊)) = (𝐵 ∩ (𝐴 ∩ (Base‘𝑊)))
11	4, 5	ressbas 17186	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑋 → (𝐴 ∩ (Base‘𝑊)) = (Base‘(𝑊 ↾_s 𝐴)))
12	3, 11	syl 17	. . . . . . . . . . 11 ⊢ (((¬ (Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝐴 ∩ (Base‘𝑊)) = (Base‘(𝑊 ↾_s 𝐴)))
13	12	ineq2d 4212	. . . . . . . . . 10 ⊢ (((¬ (Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝐵 ∩ (𝐴 ∩ (Base‘𝑊))) = (𝐵 ∩ (Base‘(𝑊 ↾_s 𝐴))))
14	10, 13	eqtr2id 2784	. . . . . . . . 9 ⊢ (((¬ (Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝐵 ∩ (Base‘(𝑊 ↾_s 𝐴))) = ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊)))
15	14	opeq2d 4880	. . . . . . . 8 ⊢ (((¬ (Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ⟨(Base‘ndx), (𝐵 ∩ (Base‘(𝑊 ↾_s 𝐴)))⟩ = ⟨(Base‘ndx), ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))⟩)
16	7, 15	oveq12d 7430	. . . . . . 7 ⊢ (((¬ (Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ((𝑊 ↾_s 𝐴) sSet ⟨(Base‘ndx), (𝐵 ∩ (Base‘(𝑊 ↾_s 𝐴)))⟩) = ((𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩) sSet ⟨(Base‘ndx), ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))⟩))
17		fvex 6904	. . . . . . . . 9 ⊢ (Base‘𝑊) ∈ V
18	17	inex2 5318	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊)) ∈ V
19		setsabs 17119	. . . . . . . 8 ⊢ ((𝑊 ∈ V ∧ ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊)) ∈ V) → ((𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩) sSet ⟨(Base‘ndx), ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))⟩) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))⟩))
20	2, 18, 19	sylancl 585	. . . . . . 7 ⊢ (((¬ (Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ((𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩) sSet ⟨(Base‘ndx), ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))⟩) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))⟩))
21	16, 20	eqtrd 2771	. . . . . 6 ⊢ (((¬ (Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ((𝑊 ↾_s 𝐴) sSet ⟨(Base‘ndx), (𝐵 ∩ (Base‘(𝑊 ↾_s 𝐴)))⟩) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))⟩))
22		simpll 764	. . . . . . 7 ⊢ (((¬ (Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ¬ (Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵)
23		ovexd 7447	. . . . . . 7 ⊢ (((¬ (Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑊 ↾_s 𝐴) ∈ V)
24		simpr3 1195	. . . . . . 7 ⊢ (((¬ (Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → 𝐵 ∈ 𝑌)
25		eqid 2731	. . . . . . . 8 ⊢ ((𝑊 ↾_s 𝐴) ↾_s 𝐵) = ((𝑊 ↾_s 𝐴) ↾_s 𝐵)
26		eqid 2731	. . . . . . . 8 ⊢ (Base‘(𝑊 ↾_s 𝐴)) = (Base‘(𝑊 ↾_s 𝐴))
27	25, 26	ressval2 17185	. . . . . . 7 ⊢ ((¬ (Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ↾_s 𝐴) ∈ V ∧ 𝐵 ∈ 𝑌) → ((𝑊 ↾_s 𝐴) ↾_s 𝐵) = ((𝑊 ↾_s 𝐴) sSet ⟨(Base‘ndx), (𝐵 ∩ (Base‘(𝑊 ↾_s 𝐴)))⟩))
28	22, 23, 24, 27	syl3anc 1370	. . . . . 6 ⊢ (((¬ (Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ((𝑊 ↾_s 𝐴) ↾_s 𝐵) = ((𝑊 ↾_s 𝐴) sSet ⟨(Base‘ndx), (𝐵 ∩ (Base‘(𝑊 ↾_s 𝐴)))⟩))
29		inss1 4228	. . . . . . . . 9 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
30		sstr 3990	. . . . . . . . 9 ⊢ (((Base‘𝑊) ⊆ (𝐴 ∩ 𝐵) ∧ (𝐴 ∩ 𝐵) ⊆ 𝐴) → (Base‘𝑊) ⊆ 𝐴)
31	29, 30	mpan2 688	. . . . . . . 8 ⊢ ((Base‘𝑊) ⊆ (𝐴 ∩ 𝐵) → (Base‘𝑊) ⊆ 𝐴)
32	1, 31	nsyl 140	. . . . . . 7 ⊢ (((¬ (Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ¬ (Base‘𝑊) ⊆ (𝐴 ∩ 𝐵))
33		inex1g 5319	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑋 → (𝐴 ∩ 𝐵) ∈ V)
34	3, 33	syl 17	. . . . . . 7 ⊢ (((¬ (Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝐴 ∩ 𝐵) ∈ V)
35		eqid 2731	. . . . . . . 8 ⊢ (𝑊 ↾_s (𝐴 ∩ 𝐵)) = (𝑊 ↾_s (𝐴 ∩ 𝐵))
36	35, 5	ressval2 17185	. . . . . . 7 ⊢ ((¬ (Base‘𝑊) ⊆ (𝐴 ∩ 𝐵) ∧ 𝑊 ∈ V ∧ (𝐴 ∩ 𝐵) ∈ V) → (𝑊 ↾_s (𝐴 ∩ 𝐵)) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))⟩))
37	32, 2, 34, 36	syl3anc 1370	. . . . . 6 ⊢ (((¬ (Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑊 ↾_s (𝐴 ∩ 𝐵)) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))⟩))
38	21, 28, 37	3eqtr4d 2781	. . . . 5 ⊢ (((¬ (Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ((𝑊 ↾_s 𝐴) ↾_s 𝐵) = (𝑊 ↾_s (𝐴 ∩ 𝐵)))
39	38	exp31 419	. . . 4 ⊢ (¬ (Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 → (¬ (Base‘𝑊) ⊆ 𝐴 → ((𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → ((𝑊 ↾_s 𝐴) ↾_s 𝐵) = (𝑊 ↾_s (𝐴 ∩ 𝐵)))))
40		ovex 7445	. . . . . . . 8 ⊢ (𝑊 ↾_s 𝐴) ∈ V
41	25, 26	ressid2 17184	. . . . . . . 8 ⊢ (((Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ↾_s 𝐴) ∈ V ∧ 𝐵 ∈ 𝑌) → ((𝑊 ↾_s 𝐴) ↾_s 𝐵) = (𝑊 ↾_s 𝐴))
42	40, 41	mp3an2 1448	. . . . . . 7 ⊢ (((Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ 𝐵 ∈ 𝑌) → ((𝑊 ↾_s 𝐴) ↾_s 𝐵) = (𝑊 ↾_s 𝐴))
43	42	3ad2antr3 1189	. . . . . 6 ⊢ (((Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ((𝑊 ↾_s 𝐴) ↾_s 𝐵) = (𝑊 ↾_s 𝐴))
44		in32 4221	. . . . . . . . 9 ⊢ ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊)) = ((𝐴 ∩ (Base‘𝑊)) ∩ 𝐵)
45		simpr2 1194	. . . . . . . . . . . 12 ⊢ (((Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → 𝐴 ∈ 𝑋)
46	45, 11	syl 17	. . . . . . . . . . 11 ⊢ (((Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝐴 ∩ (Base‘𝑊)) = (Base‘(𝑊 ↾_s 𝐴)))
47		simpl 482	. . . . . . . . . . 11 ⊢ (((Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵)
48	46, 47	eqsstrd 4020	. . . . . . . . . 10 ⊢ (((Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝐴 ∩ (Base‘𝑊)) ⊆ 𝐵)
49		df-ss 3965	. . . . . . . . . 10 ⊢ ((𝐴 ∩ (Base‘𝑊)) ⊆ 𝐵 ↔ ((𝐴 ∩ (Base‘𝑊)) ∩ 𝐵) = (𝐴 ∩ (Base‘𝑊)))
50	48, 49	sylib 217	. . . . . . . . 9 ⊢ (((Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ((𝐴 ∩ (Base‘𝑊)) ∩ 𝐵) = (𝐴 ∩ (Base‘𝑊)))
51	44, 50	eqtr2id 2784	. . . . . . . 8 ⊢ (((Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝐴 ∩ (Base‘𝑊)) = ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊)))
52	51	oveq2d 7428	. . . . . . 7 ⊢ (((Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑊 ↾_s (𝐴 ∩ (Base‘𝑊))) = (𝑊 ↾_s ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))))
53	5	ressinbas 17197	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑋 → (𝑊 ↾_s 𝐴) = (𝑊 ↾_s (𝐴 ∩ (Base‘𝑊))))
54	45, 53	syl 17	. . . . . . 7 ⊢ (((Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑊 ↾_s 𝐴) = (𝑊 ↾_s (𝐴 ∩ (Base‘𝑊))))
55	5	ressinbas 17197	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐵) ∈ V → (𝑊 ↾_s (𝐴 ∩ 𝐵)) = (𝑊 ↾_s ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))))
56	45, 33, 55	3syl 18	. . . . . . 7 ⊢ (((Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑊 ↾_s (𝐴 ∩ 𝐵)) = (𝑊 ↾_s ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))))
57	52, 54, 56	3eqtr4d 2781	. . . . . 6 ⊢ (((Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑊 ↾_s 𝐴) = (𝑊 ↾_s (𝐴 ∩ 𝐵)))
58	43, 57	eqtrd 2771	. . . . 5 ⊢ (((Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ((𝑊 ↾_s 𝐴) ↾_s 𝐵) = (𝑊 ↾_s (𝐴 ∩ 𝐵)))
59	58	ex 412	. . . 4 ⊢ ((Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 → ((𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → ((𝑊 ↾_s 𝐴) ↾_s 𝐵) = (𝑊 ↾_s (𝐴 ∩ 𝐵))))
60	4, 5	ressid2 17184	. . . . . . . 8 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑋) → (𝑊 ↾_s 𝐴) = 𝑊)
61	60	3adant3r3 1183	. . . . . . 7 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑊 ↾_s 𝐴) = 𝑊)
62	61	oveq1d 7427	. . . . . 6 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ((𝑊 ↾_s 𝐴) ↾_s 𝐵) = (𝑊 ↾_s 𝐵))
63		inss2 4229	. . . . . . . . . . 11 ⊢ (𝐵 ∩ (Base‘𝑊)) ⊆ (Base‘𝑊)
64		simpl 482	. . . . . . . . . . 11 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (Base‘𝑊) ⊆ 𝐴)
65	63, 64	sstrid 3993	. . . . . . . . . 10 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝐵 ∩ (Base‘𝑊)) ⊆ 𝐴)
66		sseqin2 4215	. . . . . . . . . 10 ⊢ ((𝐵 ∩ (Base‘𝑊)) ⊆ 𝐴 ↔ (𝐴 ∩ (𝐵 ∩ (Base‘𝑊))) = (𝐵 ∩ (Base‘𝑊)))
67	65, 66	sylib 217	. . . . . . . . 9 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝐴 ∩ (𝐵 ∩ (Base‘𝑊))) = (𝐵 ∩ (Base‘𝑊)))
68	8, 67	eqtr2id 2784	. . . . . . . 8 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝐵 ∩ (Base‘𝑊)) = ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊)))
69	68	oveq2d 7428	. . . . . . 7 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑊 ↾_s (𝐵 ∩ (Base‘𝑊))) = (𝑊 ↾_s ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))))
70		simpr3 1195	. . . . . . . 8 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → 𝐵 ∈ 𝑌)
71	5	ressinbas 17197	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑌 → (𝑊 ↾_s 𝐵) = (𝑊 ↾_s (𝐵 ∩ (Base‘𝑊))))
72	70, 71	syl 17	. . . . . . 7 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑊 ↾_s 𝐵) = (𝑊 ↾_s (𝐵 ∩ (Base‘𝑊))))
73		simpr2 1194	. . . . . . . 8 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → 𝐴 ∈ 𝑋)
74	73, 33, 55	3syl 18	. . . . . . 7 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑊 ↾_s (𝐴 ∩ 𝐵)) = (𝑊 ↾_s ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))))
75	69, 72, 74	3eqtr4d 2781	. . . . . 6 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑊 ↾_s 𝐵) = (𝑊 ↾_s (𝐴 ∩ 𝐵)))
76	62, 75	eqtrd 2771	. . . . 5 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ((𝑊 ↾_s 𝐴) ↾_s 𝐵) = (𝑊 ↾_s (𝐴 ∩ 𝐵)))
77	76	ex 412	. . . 4 ⊢ ((Base‘𝑊) ⊆ 𝐴 → ((𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → ((𝑊 ↾_s 𝐴) ↾_s 𝐵) = (𝑊 ↾_s (𝐴 ∩ 𝐵))))
78	39, 59, 77	pm2.61ii 183	. . 3 ⊢ ((𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → ((𝑊 ↾_s 𝐴) ↾_s 𝐵) = (𝑊 ↾_s (𝐴 ∩ 𝐵)))
79	78	3expib 1121	. 2 ⊢ (𝑊 ∈ V → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → ((𝑊 ↾_s 𝐴) ↾_s 𝐵) = (𝑊 ↾_s (𝐴 ∩ 𝐵))))
80		ress0 17195	. . . 4 ⊢ (∅ ↾_s 𝐵) = ∅
81		reldmress 17182	. . . . . 6 ⊢ Rel dom ↾_s
82	81	ovprc1 7451	. . . . 5 ⊢ (¬ 𝑊 ∈ V → (𝑊 ↾_s 𝐴) = ∅)
83	82	oveq1d 7427	. . . 4 ⊢ (¬ 𝑊 ∈ V → ((𝑊 ↾_s 𝐴) ↾_s 𝐵) = (∅ ↾_s 𝐵))
84	81	ovprc1 7451	. . . 4 ⊢ (¬ 𝑊 ∈ V → (𝑊 ↾_s (𝐴 ∩ 𝐵)) = ∅)
85	80, 83, 84	3eqtr4a 2797	. . 3 ⊢ (¬ 𝑊 ∈ V → ((𝑊 ↾_s 𝐴) ↾_s 𝐵) = (𝑊 ↾_s (𝐴 ∩ 𝐵)))
86	85	a1d 25	. 2 ⊢ (¬ 𝑊 ∈ V → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → ((𝑊 ↾_s 𝐴) ↾_s 𝐵) = (𝑊 ↾_s (𝐴 ∩ 𝐵))))
87	79, 86	pm2.61i 182	1 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → ((𝑊 ↾_s 𝐴) ↾_s 𝐵) = (𝑊 ↾_s (𝐴 ∩ 𝐵)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 Vcvv 3473 ∩ cin 3947 ⊆ wss 3948 ∅c0 4322 ⟨cop 4634 ‘cfv 6543 (class class class)co 7412 sSet csts 17103 ndxcnx 17133 Basecbs 17151 ↾_s cress 17180

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-1cn 11174 ax-addcl 11176

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-nn 12220 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181

This theorem is referenced by: ressabs 17201 xrge00 32621 xrge0slmod 32900 fldexttr 33192 esumpfinvallem 33537 lmhmlnmsplit 42294

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