Theorem ressress 17217

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 769	. . . . . . . . 9 ⊢ (((¬ (Base‘(𝑊 ↾s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ¬ (Base‘𝑊) ⊆ 𝐴)
2		simpr1 1196	. . . . . . . . 9 ⊢ (((¬ (Base‘(𝑊 ↾s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → 𝑊 ∈ V)
3		simpr2 1197	. . . . . . . . 9 ⊢ (((¬ (Base‘(𝑊 ↾s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → 𝐴 ∈ 𝑋)
4		eqid 2736	. . . . . . . . . 10 ⊢ (𝑊 ↾s 𝐴) = (𝑊 ↾s 𝐴)
5		eqid 2736	. . . . . . . . . 10 ⊢ (Base‘𝑊) = (Base‘𝑊)
6	4, 5	ressval2 17205	. . . . . . . . 9 ⊢ ((¬ (Base‘𝑊) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑋) → (𝑊 ↾s 𝐴) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩))
7	1, 2, 3, 6	syl3anc 1374	. . . . . . . 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 4168	. . . . . . . . . . 11 ⊢ ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊)) = (𝐴 ∩ (𝐵 ∩ (Base‘𝑊)))
9		in12 4169	. . . . . . . . . . 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 17206	. . . . . . . . . . . 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 4160	. . . . . . . . . 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 4823	. . . . . . . 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 7385	. . . . . . 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 6853	. . . . . . . . 9 ⊢ (Base‘𝑊) ∈ V
18	17	inex2 5259	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊)) ∈ V
19		setsabs 17149	. . . . . . . 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 587	. . . . . . 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 767	. . . . . . 7 ⊢ (((¬ (Base‘(𝑊 ↾s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ¬ (Base‘(𝑊 ↾s 𝐴)) ⊆ 𝐵)
23		ovexd 7402	. . . . . . 7 ⊢ (((¬ (Base‘(𝑊 ↾s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑊 ↾s 𝐴) ∈ V)
24		simpr3 1198	. . . . . . 7 ⊢ (((¬ (Base‘(𝑊 ↾s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → 𝐵 ∈ 𝑌)
25		eqid 2736	. . . . . . . 8 ⊢ ((𝑊 ↾s 𝐴) ↾s 𝐵) = ((𝑊 ↾s 𝐴) ↾s 𝐵)
26		eqid 2736	. . . . . . . 8 ⊢ (Base‘(𝑊 ↾s 𝐴)) = (Base‘(𝑊 ↾s 𝐴))
27	25, 26	ressval2 17205	. . . . . . 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 1374	. . . . . 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 4177	. . . . . . . . 9 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
30		sstr 3930	. . . . . . . . 9 ⊢ (((Base‘𝑊) ⊆ (𝐴 ∩ 𝐵) ∧ (𝐴 ∩ 𝐵) ⊆ 𝐴) → (Base‘𝑊) ⊆ 𝐴)
31	29, 30	mpan2 692	. . . . . . . 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 5260	. . . . . . . 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 2736	. . . . . . . 8 ⊢ (𝑊 ↾s (𝐴 ∩ 𝐵)) = (𝑊 ↾s (𝐴 ∩ 𝐵))
36	35, 5	ressval2 17205	. . . . . . 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 1374	. . . . . 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 7400	. . . . . . . 8 ⊢ (𝑊 ↾s 𝐴) ∈ V
41	25, 26	ressid2 17204	. . . . . . . 8 ⊢ (((Base‘(𝑊 ↾s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ↾s 𝐴) ∈ V ∧ 𝐵 ∈ 𝑌) → ((𝑊 ↾s 𝐴) ↾s 𝐵) = (𝑊 ↾s 𝐴))
42	40, 41	mp3an2 1452	. . . . . . 7 ⊢ (((Base‘(𝑊 ↾s 𝐴)) ⊆ 𝐵 ∧ 𝐵 ∈ 𝑌) → ((𝑊 ↾s 𝐴) ↾s 𝐵) = (𝑊 ↾s 𝐴))
43	42	3ad2antr3 1192	. . . . . 6 ⊢ (((Base‘(𝑊 ↾s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ((𝑊 ↾s 𝐴) ↾s 𝐵) = (𝑊 ↾s 𝐴))
44		in32 4170	. . . . . . . . 9 ⊢ ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊)) = ((𝐴 ∩ (Base‘𝑊)) ∩ 𝐵)
45		simpr2 1197	. . . . . . . . . . . 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 3956	. . . . . . . . . 10 ⊢ (((Base‘(𝑊 ↾s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝐴 ∩ (Base‘𝑊)) ⊆ 𝐵)
49		dfss2 3907	. . . . . . . . . 10 ⊢ ((𝐴 ∩ (Base‘𝑊)) ⊆ 𝐵 ↔ ((𝐴 ∩ (Base‘𝑊)) ∩ 𝐵) = (𝐴 ∩ (Base‘𝑊)))
50	48, 49	sylib 218	. . . . . . . . 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 7383	. . . . . . 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 17215	. . . . . . . 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 17215	. . . . . . . 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 17204	. . . . . . . 8 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑋) → (𝑊 ↾s 𝐴) = 𝑊)
61	60	3adant3r3 1186	. . . . . . 7 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑊 ↾s 𝐴) = 𝑊)
62	61	oveq1d 7382	. . . . . 6 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ((𝑊 ↾s 𝐴) ↾s 𝐵) = (𝑊 ↾s 𝐵))
63		inss2 4178	. . . . . . . . . . 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 3933	. . . . . . . . . 10 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝐵 ∩ (Base‘𝑊)) ⊆ 𝐴)
66		sseqin2 4163	. . . . . . . . . 10 ⊢ ((𝐵 ∩ (Base‘𝑊)) ⊆ 𝐴 ↔ (𝐴 ∩ (𝐵 ∩ (Base‘𝑊))) = (𝐵 ∩ (Base‘𝑊)))
67	65, 66	sylib 218	. . . . . . . . 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 7383	. . . . . . 7 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑊 ↾s (𝐵 ∩ (Base‘𝑊))) = (𝑊 ↾s ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))))
70		simpr3 1198	. . . . . . . 8 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → 𝐵 ∈ 𝑌)
71	5	ressinbas 17215	. . . . . . . 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 1197	. . . . . . . 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 1123	. 2 ⊢ (𝑊 ∈ V → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → ((𝑊 ↾s 𝐴) ↾s 𝐵) = (𝑊 ↾s (𝐴 ∩ 𝐵))))
80		ress0 17213	. . . 4 ⊢ (∅ ↾s 𝐵) = ∅
81		reldmress 17202	. . . . . 6 ⊢ Rel dom ↾s
82	81	ovprc1 7406	. . . . 5 ⊢ (¬ 𝑊 ∈ V → (𝑊 ↾s 𝐴) = ∅)
83	82	oveq1d 7382	. . . 4 ⊢ (¬ 𝑊 ∈ V → ((𝑊 ↾s 𝐴) ↾s 𝐵) = (∅ ↾s 𝐵))
84	81	ovprc1 7406	. . . 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)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  Vcvv 3429   ∩ cin 3888   ⊆ wss 3889  ∅c0 4273  ⟨cop 4573  ‘cfv 6498  (class class class)co 7367   sSet csts 17133  ndxcnx 17163  Basecbs 17179   ↾_s cress 17200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-1cn 11096  ax-addcl 11098

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-nn 12175  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201

This theorem is referenced by:  ressabs  17218  xrge00  33074  xrge0slmod  33408  fldexttr  33802  fldgenfldext  33812  esumpfinvallem  34218  lmhmlnmsplit  43515