Theorem ressress 17212

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 2737	. . . . . . . . . 10 ⊢ (𝑊 ↾s 𝐴) = (𝑊 ↾s 𝐴)
5		eqid 2737	. . . . . . . . . 10 ⊢ (Base‘𝑊) = (Base‘𝑊)
6	4, 5	ressval2 17200	. . . . . . . . 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 4169	. . . . . . . . . . 11 ⊢ ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊)) = (𝐴 ∩ (𝐵 ∩ (Base‘𝑊)))
9		in12 4170	. . . . . . . . . . 11 ⊢ (𝐴 ∩ (𝐵 ∩ (Base‘𝑊))) = (𝐵 ∩ (𝐴 ∩ (Base‘𝑊)))
10	8, 9	eqtri 2760	. . . . . . . . . 10 ⊢ ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊)) = (𝐵 ∩ (𝐴 ∩ (Base‘𝑊)))
11	4, 5	ressbas 17201	. . . . . . . . . . . 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 4161	. . . . . . . . . 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 2785	. . . . . . . . 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 4824	. . . . . . . 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 7380	. . . . . . 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 6849	. . . . . . . . 9 ⊢ (Base‘𝑊) ∈ V
18	17	inex2 5256	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊)) ∈ V
19		setsabs 17144	. . . . . . . 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 2772	. . . . . 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 7397	. . . . . . 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 2737	. . . . . . . 8 ⊢ ((𝑊 ↾s 𝐴) ↾s 𝐵) = ((𝑊 ↾s 𝐴) ↾s 𝐵)
26		eqid 2737	. . . . . . . 8 ⊢ (Base‘(𝑊 ↾s 𝐴)) = (Base‘(𝑊 ↾s 𝐴))
27	25, 26	ressval2 17200	. . . . . . 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 4178	. . . . . . . . 9 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
30		sstr 3931	. . . . . . . . 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 5257	. . . . . . . 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 2737	. . . . . . . 8 ⊢ (𝑊 ↾s (𝐴 ∩ 𝐵)) = (𝑊 ↾s (𝐴 ∩ 𝐵))
36	35, 5	ressval2 17200	. . . . . . 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 2782	. . . . 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 7395	. . . . . . . 8 ⊢ (𝑊 ↾s 𝐴) ∈ V
41	25, 26	ressid2 17199	. . . . . . . 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 4171	. . . . . . . . 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 3957	. . . . . . . . . 10 ⊢ (((Base‘(𝑊 ↾s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝐴 ∩ (Base‘𝑊)) ⊆ 𝐵)
49		dfss2 3908	. . . . . . . . . 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 2785	. . . . . . . 8 ⊢ (((Base‘(𝑊 ↾s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝐴 ∩ (Base‘𝑊)) = ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊)))
52	51	oveq2d 7378	. . . . . . 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 17210	. . . . . . . 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 17210	. . . . . . . 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 2782	. . . . . 6 ⊢ (((Base‘(𝑊 ↾s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑊 ↾s 𝐴) = (𝑊 ↾s (𝐴 ∩ 𝐵)))
58	43, 57	eqtrd 2772	. . . . 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 17199	. . . . . . . 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 7377	. . . . . 6 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ((𝑊 ↾s 𝐴) ↾s 𝐵) = (𝑊 ↾s 𝐵))
63		inss2 4179	. . . . . . . . . . 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 3934	. . . . . . . . . 10 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝐵 ∩ (Base‘𝑊)) ⊆ 𝐴)
66		sseqin2 4164	. . . . . . . . . 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 2785	. . . . . . . 8 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝐵 ∩ (Base‘𝑊)) = ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊)))
69	68	oveq2d 7378	. . . . . . 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 17210	. . . . . . . 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 2782	. . . . . 6 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑊 ↾s 𝐵) = (𝑊 ↾s (𝐴 ∩ 𝐵)))
76	62, 75	eqtrd 2772	. . . . 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 17208	. . . 4 ⊢ (∅ ↾s 𝐵) = ∅
81		reldmress 17197	. . . . . 6 ⊢ Rel dom ↾s
82	81	ovprc1 7401	. . . . 5 ⊢ (¬ 𝑊 ∈ V → (𝑊 ↾s 𝐴) = ∅)
83	82	oveq1d 7377	. . . 4 ⊢ (¬ 𝑊 ∈ V → ((𝑊 ↾s 𝐴) ↾s 𝐵) = (∅ ↾s 𝐵))
84	81	ovprc1 7401	. . . 4 ⊢ (¬ 𝑊 ∈ V → (𝑊 ↾s (𝐴 ∩ 𝐵)) = ∅)
85	80, 83, 84	3eqtr4a 2798	. . 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 3430   ∩ cin 3889   ⊆ wss 3890  ∅c0 4274  ⟨cop 4574  ‘cfv 6494  (class class class)co 7362   sSet csts 17128  ndxcnx 17158  Basecbs 17174   ↾_s cress 17195

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 2709  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684  ax-cnex 11089  ax-1cn 11091  ax-addcl 11093

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-pred 6261  df-ord 6322  df-on 6323  df-lim 6324  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7813  df-2nd 7938  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-nn 12170  df-sets 17129  df-slot 17147  df-ndx 17159  df-base 17175  df-ress 17196

This theorem is referenced by:  ressabs  17213  xrge00  33093  xrge0slmod  33427  fldexttr  33822  fldgenfldext  33832  esumpfinvallem  34238  lmhmlnmsplit  43537