Theorem ressress 17361

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 781	. . . . . . . . 9 ⊢ (((¬ (Base‘(𝑊 ↾s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ¬ (Base‘𝑊) ⊆ 𝐴)
2		simpr1 1213	. . . . . . . . 9 ⊢ (((¬ (Base‘(𝑊 ↾s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → 𝑊 ∈ V)
3		simpr2 1214	. . . . . . . . 9 ⊢ (((¬ (Base‘(𝑊 ↾s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → 𝐴 ∈ 𝑋)
4		eqid 2760	. . . . . . . . . 10 ⊢ (𝑊 ↾s 𝐴) = (𝑊 ↾s 𝐴)
5		eqid 2760	. . . . . . . . . 10 ⊢ (Base‘𝑊) = (Base‘𝑊)
6	4, 5	ressval2 17349	. . . . . . . . 9 ⊢ ((¬ (Base‘𝑊) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑋) → (𝑊 ↾s 𝐴) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩))
7	1, 2, 3, 6	syl3anc 1398	. . . . . . . 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 4173	. . . . . . . . . . 11 ⊢ ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊)) = (𝐴 ∩ (𝐵 ∩ (Base‘𝑊)))
9		in12 4174	. . . . . . . . . . 11 ⊢ (𝐴 ∩ (𝐵 ∩ (Base‘𝑊))) = (𝐵 ∩ (𝐴 ∩ (Base‘𝑊)))
10	8, 9	eqtri 2783	. . . . . . . . . 10 ⊢ ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊)) = (𝐵 ∩ (𝐴 ∩ (Base‘𝑊)))
11	4, 5	ressbas 17350	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑋 → (𝐴 ∩ (Base‘𝑊)) = (Base‘(𝑊 ↾s 𝐴)))
12	3, 11	syl 18	. . . . . . . . . . 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 4166	. . . . . . . . . 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 2808	. . . . . . . . 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 4840	. . . . . . . 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 7433	. . . . . . 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 6893	. . . . . . . . 9 ⊢ (Base‘𝑊) ∈ V
18	17	inex2 5281	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊)) ∈ V
19		setsabs 17293	. . . . . . . 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 598	. . . . . . 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 2795	. . . . . 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 779	. . . . . . 7 ⊢ (((¬ (Base‘(𝑊 ↾s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ¬ (Base‘(𝑊 ↾s 𝐴)) ⊆ 𝐵)
23		ovexd 7450	. . . . . . 7 ⊢ (((¬ (Base‘(𝑊 ↾s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑊 ↾s 𝐴) ∈ V)
24		simpr3 1215	. . . . . . 7 ⊢ (((¬ (Base‘(𝑊 ↾s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → 𝐵 ∈ 𝑌)
25		eqid 2760	. . . . . . . 8 ⊢ ((𝑊 ↾s 𝐴) ↾s 𝐵) = ((𝑊 ↾s 𝐴) ↾s 𝐵)
26		eqid 2760	. . . . . . . 8 ⊢ (Base‘(𝑊 ↾s 𝐴)) = (Base‘(𝑊 ↾s 𝐴))
27	25, 26	ressval2 17349	. . . . . . 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 1398	. . . . . 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 4182	. . . . . . . . 9 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
30		sstr 3939	. . . . . . . . 9 ⊢ (((Base‘𝑊) ⊆ (𝐴 ∩ 𝐵) ∧ (𝐴 ∩ 𝐵) ⊆ 𝐴) → (Base‘𝑊) ⊆ 𝐴)
31	29, 30	mpan2 704	. . . . . . . 8 ⊢ ((Base‘𝑊) ⊆ (𝐴 ∩ 𝐵) → (Base‘𝑊) ⊆ 𝐴)
32	1, 31	nsyl 141	. . . . . . 7 ⊢ (((¬ (Base‘(𝑊 ↾s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ¬ (Base‘𝑊) ⊆ (𝐴 ∩ 𝐵))
33		inex1g 5282	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑋 → (𝐴 ∩ 𝐵) ∈ V)
34	3, 33	syl 18	. . . . . . 7 ⊢ (((¬ (Base‘(𝑊 ↾s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝐴 ∩ 𝐵) ∈ V)
35		eqid 2760	. . . . . . . 8 ⊢ (𝑊 ↾s (𝐴 ∩ 𝐵)) = (𝑊 ↾s (𝐴 ∩ 𝐵))
36	35, 5	ressval2 17349	. . . . . . 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 1398	. . . . . 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 2805	. . . . 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 425	. . . 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 7448	. . . . . . . 8 ⊢ (𝑊 ↾s 𝐴) ∈ V
41	25, 26	ressid2 17348	. . . . . . . 8 ⊢ (((Base‘(𝑊 ↾s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ↾s 𝐴) ∈ V ∧ 𝐵 ∈ 𝑌) → ((𝑊 ↾s 𝐴) ↾s 𝐵) = (𝑊 ↾s 𝐴))
42	40, 41	mp3an2 1478	. . . . . . 7 ⊢ (((Base‘(𝑊 ↾s 𝐴)) ⊆ 𝐵 ∧ 𝐵 ∈ 𝑌) → ((𝑊 ↾s 𝐴) ↾s 𝐵) = (𝑊 ↾s 𝐴))
43	42	3ad2antr3 1209	. . . . . 6 ⊢ (((Base‘(𝑊 ↾s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ((𝑊 ↾s 𝐴) ↾s 𝐵) = (𝑊 ↾s 𝐴))
44		in32 4175	. . . . . . . . 9 ⊢ ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊)) = ((𝐴 ∩ (Base‘𝑊)) ∩ 𝐵)
45		simpr2 1214	. . . . . . . . . . . 12 ⊢ (((Base‘(𝑊 ↾s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → 𝐴 ∈ 𝑋)
46	45, 11	syl 18	. . . . . . . . . . 11 ⊢ (((Base‘(𝑊 ↾s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝐴 ∩ (Base‘𝑊)) = (Base‘(𝑊 ↾s 𝐴)))
47		simpl 488	. . . . . . . . . . 11 ⊢ (((Base‘(𝑊 ↾s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (Base‘(𝑊 ↾s 𝐴)) ⊆ 𝐵)
48	46, 47	eqsstrd 3965	. . . . . . . . . 10 ⊢ (((Base‘(𝑊 ↾s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝐴 ∩ (Base‘𝑊)) ⊆ 𝐵)
49		dfss2 3917	. . . . . . . . . 10 ⊢ ((𝐴 ∩ (Base‘𝑊)) ⊆ 𝐵 ↔ ((𝐴 ∩ (Base‘𝑊)) ∩ 𝐵) = (𝐴 ∩ (Base‘𝑊)))
50	48, 49	sylib 221	. . . . . . . . 9 ⊢ (((Base‘(𝑊 ↾s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ((𝐴 ∩ (Base‘𝑊)) ∩ 𝐵) = (𝐴 ∩ (Base‘𝑊)))
51	44, 50	eqtr2id 2808	. . . . . . . 8 ⊢ (((Base‘(𝑊 ↾s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝐴 ∩ (Base‘𝑊)) = ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊)))
52	51	oveq2d 7431	. . . . . . 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 17359	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑋 → (𝑊 ↾s 𝐴) = (𝑊 ↾s (𝐴 ∩ (Base‘𝑊))))
54	45, 53	syl 18	. . . . . . 7 ⊢ (((Base‘(𝑊 ↾s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑊 ↾s 𝐴) = (𝑊 ↾s (𝐴 ∩ (Base‘𝑊))))
55	5	ressinbas 17359	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐵) ∈ V → (𝑊 ↾s (𝐴 ∩ 𝐵)) = (𝑊 ↾s ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))))
56	45, 33, 55	3syl 19	. . . . . . 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 2805	. . . . . 6 ⊢ (((Base‘(𝑊 ↾s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑊 ↾s 𝐴) = (𝑊 ↾s (𝐴 ∩ 𝐵)))
58	43, 57	eqtrd 2795	. . . . 5 ⊢ (((Base‘(𝑊 ↾s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ((𝑊 ↾s 𝐴) ↾s 𝐵) = (𝑊 ↾s (𝐴 ∩ 𝐵)))
59	58	ex 418	. . . 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 17348	. . . . . . . 8 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑋) → (𝑊 ↾s 𝐴) = 𝑊)
61	60	3adant3r3 1203	. . . . . . 7 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑊 ↾s 𝐴) = 𝑊)
62	61	oveq1d 7430	. . . . . 6 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ((𝑊 ↾s 𝐴) ↾s 𝐵) = (𝑊 ↾s 𝐵))
63		inss2 4183	. . . . . . . . . . 11 ⊢ (𝐵 ∩ (Base‘𝑊)) ⊆ (Base‘𝑊)
64		simpl 488	. . . . . . . . . . 11 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (Base‘𝑊) ⊆ 𝐴)
65	63, 64	sstrid 3942	. . . . . . . . . 10 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝐵 ∩ (Base‘𝑊)) ⊆ 𝐴)
66		sseqin2 4169	. . . . . . . . . 10 ⊢ ((𝐵 ∩ (Base‘𝑊)) ⊆ 𝐴 ↔ (𝐴 ∩ (𝐵 ∩ (Base‘𝑊))) = (𝐵 ∩ (Base‘𝑊)))
67	65, 66	sylib 221	. . . . . . . . 9 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝐴 ∩ (𝐵 ∩ (Base‘𝑊))) = (𝐵 ∩ (Base‘𝑊)))
68	8, 67	eqtr2id 2808	. . . . . . . 8 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝐵 ∩ (Base‘𝑊)) = ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊)))
69	68	oveq2d 7431	. . . . . . 7 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑊 ↾s (𝐵 ∩ (Base‘𝑊))) = (𝑊 ↾s ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))))
70		simpr3 1215	. . . . . . . 8 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → 𝐵 ∈ 𝑌)
71	5	ressinbas 17359	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑌 → (𝑊 ↾s 𝐵) = (𝑊 ↾s (𝐵 ∩ (Base‘𝑊))))
72	70, 71	syl 18	. . . . . . 7 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑊 ↾s 𝐵) = (𝑊 ↾s (𝐵 ∩ (Base‘𝑊))))
73		simpr2 1214	. . . . . . . 8 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → 𝐴 ∈ 𝑋)
74	73, 33, 55	3syl 19	. . . . . . 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 2805	. . . . . 6 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑊 ↾s 𝐵) = (𝑊 ↾s (𝐴 ∩ 𝐵)))
76	62, 75	eqtrd 2795	. . . . 5 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ((𝑊 ↾s 𝐴) ↾s 𝐵) = (𝑊 ↾s (𝐴 ∩ 𝐵)))
77	76	ex 418	. . . 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 185	. . 3 ⊢ ((𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → ((𝑊 ↾s 𝐴) ↾s 𝐵) = (𝑊 ↾s (𝐴 ∩ 𝐵)))
79	78	3expib 1140	. 2 ⊢ (𝑊 ∈ V → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → ((𝑊 ↾s 𝐴) ↾s 𝐵) = (𝑊 ↾s (𝐴 ∩ 𝐵))))
80		ress0 17357	. . . 4 ⊢ (∅ ↾s 𝐵) = ∅
81		reldmress 17346	. . . . . 6 ⊢ Rel dom ↾s
82	81	ovprc1 7454	. . . . 5 ⊢ (¬ 𝑊 ∈ V → (𝑊 ↾s 𝐴) = ∅)
83	82	oveq1d 7430	. . . 4 ⊢ (¬ 𝑊 ∈ V → ((𝑊 ↾s 𝐴) ↾s 𝐵) = (∅ ↾s 𝐵))
84	81	ovprc1 7454	. . . 4 ⊢ (¬ 𝑊 ∈ V → (𝑊 ↾s (𝐴 ∩ 𝐵)) = ∅)
85	80, 83, 84	3eqtr4a 2821	. . 3 ⊢ (¬ 𝑊 ∈ V → ((𝑊 ↾s 𝐴) ↾s 𝐵) = (𝑊 ↾s (𝐴 ∩ 𝐵)))
86	85	a1d 26	. 2 ⊢ (¬ 𝑊 ∈ V → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → ((𝑊 ↾s 𝐴) ↾s 𝐵) = (𝑊 ↾s (𝐴 ∩ 𝐵))))
87	79, 86	pm2.61i 184	1 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → ((𝑊 ↾s 𝐴) ↾s 𝐵) = (𝑊 ↾s (𝐴 ∩ 𝐵)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  Vcvv 3450   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279  ⟨cop 4590  ‘cfv 6534  (class class class)co 7415   sSet csts 17277  ndxcnx 17307  Basecbs 17323   ↾_s cress 17344

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7738  ax-cnex 11202  ax-1cn 11204  ax-addcl 11206

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6300  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7418  df-oprab 7419  df-mpo 7420  df-om 7865  df-2nd 7989  df-frecs 8282  df-wrecs 8313  df-recs 8362  df-rdg 8401  df-nn 12280  df-sets 17278  df-slot 17296  df-ndx 17308  df-base 17324  df-ress 17345

This theorem is used by:  ressabs  17362  xrge00  33483  xrge0slmod  33817  fldexttr  34198  fldgenfldext  34208  esumpfinvallem  34614  lmhmlnmsplit  43942