Theorem restntr 23480

Description: An interior in a subspace topology. Willard in General Topology says that there is no analogue of restcls 23479 for interiors. In some sense, that is true. (Contributed by Jeff Hankins, 23-Jan-2010.) (Revised by Mario Carneiro, 15-Dec-2013.)

Hypotheses

Ref	Expression
restcls.1	⊢ 𝑋 = ∪ 𝐽
restcls.2	⊢ 𝐾 = (𝐽 ↾t 𝑌)

Assertion

Ref	Expression
restntr	⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((int‘𝐾)‘𝑆) = (((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∩ 𝑌))

Proof of Theorem restntr

Dummy variables 𝑥 𝑜 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		restcls.2	. . . . . . 7 ⊢ 𝐾 = (𝐽 ↾t 𝑌)
2	1	fveq2i 6880	. . . . . 6 ⊢ (int‘𝐾) = (int‘(𝐽 ↾t 𝑌))
3	2	fveq1i 6878	. . . . 5 ⊢ ((int‘𝐾)‘𝑆) = ((int‘(𝐽 ↾t 𝑌))‘𝑆)
4		restcls.1	. . . . . . . . . 10 ⊢ 𝑋 = ∪ 𝐽
5	4	topopn 23204	. . . . . . . . 9 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
6		ssexg 5281	. . . . . . . . . 10 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑋 ∈ 𝐽) → 𝑌 ∈ V)
7	6	ancoms 464	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐽 ∧ 𝑌 ⊆ 𝑋) → 𝑌 ∈ V)
8	5, 7	sylan 592	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → 𝑌 ∈ V)
9		resttop 23458	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ∈ V) → (𝐽 ↾t 𝑌) ∈ Top)
10	8, 9	syldan 603	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → (𝐽 ↾t 𝑌) ∈ Top)
11	10	3adant3 1150	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (𝐽 ↾t 𝑌) ∈ Top)
12	4	restuni 23460	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → 𝑌 = ∪ (𝐽 ↾t 𝑌))
13	12	sseq2d 3963	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → (𝑆 ⊆ 𝑌 ↔ 𝑆 ⊆ ∪ (𝐽 ↾t 𝑌)))
14	13	biimp3a 1498	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ ∪ (𝐽 ↾t 𝑌))
15		eqid 2761	. . . . . . 7 ⊢ ∪ (𝐽 ↾t 𝑌) = ∪ (𝐽 ↾t 𝑌)
16	15	ntropn 23347	. . . . . 6 ⊢ (((𝐽 ↾t 𝑌) ∈ Top ∧ 𝑆 ⊆ ∪ (𝐽 ↾t 𝑌)) → ((int‘(𝐽 ↾t 𝑌))‘𝑆) ∈ (𝐽 ↾t 𝑌))
17	11, 14, 16	syl2anc 596	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((int‘(𝐽 ↾t 𝑌))‘𝑆) ∈ (𝐽 ↾t 𝑌))
18	3, 17	eqeltrid 2865	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((int‘𝐾)‘𝑆) ∈ (𝐽 ↾t 𝑌))
19		simp1 1154	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝐽 ∈ Top)
20		uniexg 7746	. . . . . . . . 9 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ V)
21	4, 20	eqeltrid 2865	. . . . . . . 8 ⊢ (𝐽 ∈ Top → 𝑋 ∈ V)
22		ssexg 5281	. . . . . . . 8 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑋 ∈ V) → 𝑌 ∈ V)
23	21, 22	sylan2 605	. . . . . . 7 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝐽 ∈ Top) → 𝑌 ∈ V)
24	23	ancoms 464	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → 𝑌 ∈ V)
25	24	3adant3 1150	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑌 ∈ V)
26		elrest 17578	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ∈ V) → (((int‘𝐾)‘𝑆) ∈ (𝐽 ↾t 𝑌) ↔ ∃𝑜 ∈ 𝐽 ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌)))
27	19, 25, 26	syl2anc 596	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (((int‘𝐾)‘𝑆) ∈ (𝐽 ↾t 𝑌) ↔ ∃𝑜 ∈ 𝐽 ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌)))
28	18, 27	mpbid 235	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ∃𝑜 ∈ 𝐽 ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌))
29	4	eltopss 23205	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → 𝑜 ⊆ 𝑋)
30	29	sseld 3930	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → (𝑥 ∈ 𝑜 → 𝑥 ∈ 𝑋))
31	30	adantrr 730	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ (𝑜 ∈ 𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌))) → (𝑥 ∈ 𝑜 → 𝑥 ∈ 𝑋))
32	31	3ad2antl1 1204	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌))) → (𝑥 ∈ 𝑜 → 𝑥 ∈ 𝑋))
33		eldif 3909	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑋 ∖ 𝑌) ↔ (𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ 𝑌))
34	33	simplbi2 506	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋 → (¬ 𝑥 ∈ 𝑌 → 𝑥 ∈ (𝑋 ∖ 𝑌)))
35	34	orrd 877	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑋 → (𝑥 ∈ 𝑌 ∨ 𝑥 ∈ (𝑋 ∖ 𝑌)))
36	32, 35	syl6 36	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌))) → (𝑥 ∈ 𝑜 → (𝑥 ∈ 𝑌 ∨ 𝑥 ∈ (𝑋 ∖ 𝑌))))
37		elin 3915	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝑜 ∩ 𝑌) ↔ (𝑥 ∈ 𝑜 ∧ 𝑥 ∈ 𝑌))
38		eleq2 2850	. . . . . . . . . . . . 13 ⊢ (((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌) → (𝑥 ∈ ((int‘𝐾)‘𝑆) ↔ 𝑥 ∈ (𝑜 ∩ 𝑌)))
39		elun1 4128	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ((int‘𝐾)‘𝑆) → 𝑥 ∈ (((int‘𝐾)‘𝑆) ∪ (𝑋 ∖ 𝑌)))
40	38, 39	biimtrrdi 257	. . . . . . . . . . . 12 ⊢ (((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌) → (𝑥 ∈ (𝑜 ∩ 𝑌) → 𝑥 ∈ (((int‘𝐾)‘𝑆) ∪ (𝑋 ∖ 𝑌))))
41	40	ad2antll 742	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌))) → (𝑥 ∈ (𝑜 ∩ 𝑌) → 𝑥 ∈ (((int‘𝐾)‘𝑆) ∪ (𝑋 ∖ 𝑌))))
42	37, 41	biimtrrid 246	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌))) → ((𝑥 ∈ 𝑜 ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ (((int‘𝐾)‘𝑆) ∪ (𝑋 ∖ 𝑌))))
43	42	expdimp 458	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌))) ∧ 𝑥 ∈ 𝑜) → (𝑥 ∈ 𝑌 → 𝑥 ∈ (((int‘𝐾)‘𝑆) ∪ (𝑋 ∖ 𝑌))))
44		elun2 4129	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑋 ∖ 𝑌) → 𝑥 ∈ (((int‘𝐾)‘𝑆) ∪ (𝑋 ∖ 𝑌)))
45	44	a1i 11	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌))) ∧ 𝑥 ∈ 𝑜) → (𝑥 ∈ (𝑋 ∖ 𝑌) → 𝑥 ∈ (((int‘𝐾)‘𝑆) ∪ (𝑋 ∖ 𝑌))))
46	43, 45	jaod 873	. . . . . . . 8 ⊢ ((((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌))) ∧ 𝑥 ∈ 𝑜) → ((𝑥 ∈ 𝑌 ∨ 𝑥 ∈ (𝑋 ∖ 𝑌)) → 𝑥 ∈ (((int‘𝐾)‘𝑆) ∪ (𝑋 ∖ 𝑌))))
47	46	ex 418	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌))) → (𝑥 ∈ 𝑜 → ((𝑥 ∈ 𝑌 ∨ 𝑥 ∈ (𝑋 ∖ 𝑌)) → 𝑥 ∈ (((int‘𝐾)‘𝑆) ∪ (𝑋 ∖ 𝑌)))))
48	36, 47	mpdd 44	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌))) → (𝑥 ∈ 𝑜 → 𝑥 ∈ (((int‘𝐾)‘𝑆) ∪ (𝑋 ∖ 𝑌))))
49	48	ssrdv 3937	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌))) → 𝑜 ⊆ (((int‘𝐾)‘𝑆) ∪ (𝑋 ∖ 𝑌)))
50	11	adantr 486	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌))) → (𝐽 ↾t 𝑌) ∈ Top)
51	1, 50	eqeltrid 2865	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌))) → 𝐾 ∈ Top)
52	14	adantr 486	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌))) → 𝑆 ⊆ ∪ (𝐽 ↾t 𝑌))
53	1	unieqi 4879	. . . . . . . . 9 ⊢ ∪ 𝐾 = ∪ (𝐽 ↾t 𝑌)
54	53	eqcomi 2770	. . . . . . . 8 ⊢ ∪ (𝐽 ↾t 𝑌) = ∪ 𝐾
55	54	ntrss2 23355	. . . . . . 7 ⊢ ((𝐾 ∈ Top ∧ 𝑆 ⊆ ∪ (𝐽 ↾t 𝑌)) → ((int‘𝐾)‘𝑆) ⊆ 𝑆)
56	51, 52, 55	syl2anc 596	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌))) → ((int‘𝐾)‘𝑆) ⊆ 𝑆)
57		unss1 4131	. . . . . 6 ⊢ (((int‘𝐾)‘𝑆) ⊆ 𝑆 → (((int‘𝐾)‘𝑆) ∪ (𝑋 ∖ 𝑌)) ⊆ (𝑆 ∪ (𝑋 ∖ 𝑌)))
58	56, 57	syl 18	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌))) → (((int‘𝐾)‘𝑆) ∪ (𝑋 ∖ 𝑌)) ⊆ (𝑆 ∪ (𝑋 ∖ 𝑌)))
59	49, 58	sstrd 3941	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌))) → 𝑜 ⊆ (𝑆 ∪ (𝑋 ∖ 𝑌)))
60		simpl1 1210	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ 𝑜 ⊆ (𝑆 ∪ (𝑋 ∖ 𝑌)))) → 𝐽 ∈ Top)
61		sstr 3939	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝑋) → 𝑆 ⊆ 𝑋)
62	61	ancoms 464	. . . . . . . . . . . . 13 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ 𝑋)
63	62	3adant1 1148	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ 𝑋)
64	63	adantr 486	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ 𝑜 ⊆ (𝑆 ∪ (𝑋 ∖ 𝑌)))) → 𝑆 ⊆ 𝑋)
65		difss 4083	. . . . . . . . . . 11 ⊢ (𝑋 ∖ 𝑌) ⊆ 𝑋
66		unss 4136	. . . . . . . . . . 11 ⊢ ((𝑆 ⊆ 𝑋 ∧ (𝑋 ∖ 𝑌) ⊆ 𝑋) ↔ (𝑆 ∪ (𝑋 ∖ 𝑌)) ⊆ 𝑋)
67	64, 65, 66	sylanblc 601	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ 𝑜 ⊆ (𝑆 ∪ (𝑋 ∖ 𝑌)))) → (𝑆 ∪ (𝑋 ∖ 𝑌)) ⊆ 𝑋)
68		simprl 783	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ 𝑜 ⊆ (𝑆 ∪ (𝑋 ∖ 𝑌)))) → 𝑜 ∈ 𝐽)
69		simprr 785	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ 𝑜 ⊆ (𝑆 ∪ (𝑋 ∖ 𝑌)))) → 𝑜 ⊆ (𝑆 ∪ (𝑋 ∖ 𝑌)))
70	4	ssntr 23356	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ (𝑆 ∪ (𝑋 ∖ 𝑌)) ⊆ 𝑋) ∧ (𝑜 ∈ 𝐽 ∧ 𝑜 ⊆ (𝑆 ∪ (𝑋 ∖ 𝑌)))) → 𝑜 ⊆ ((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))))
71	60, 67, 68, 69, 70	syl22anc 852	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ 𝑜 ⊆ (𝑆 ∪ (𝑋 ∖ 𝑌)))) → 𝑜 ⊆ ((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))))
72	71	ssrind 4189	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ 𝑜 ⊆ (𝑆 ∪ (𝑋 ∖ 𝑌)))) → (𝑜 ∩ 𝑌) ⊆ (((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∩ 𝑌))
73		sseq1 3956	. . . . . . . 8 ⊢ (((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌) → (((int‘𝐾)‘𝑆) ⊆ (((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∩ 𝑌) ↔ (𝑜 ∩ 𝑌) ⊆ (((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∩ 𝑌)))
74	72, 73	syl5ibrcom 250	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ 𝑜 ⊆ (𝑆 ∪ (𝑋 ∖ 𝑌)))) → (((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌) → ((int‘𝐾)‘𝑆) ⊆ (((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∩ 𝑌)))
75	74	expr 462	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ 𝑜 ∈ 𝐽) → (𝑜 ⊆ (𝑆 ∪ (𝑋 ∖ 𝑌)) → (((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌) → ((int‘𝐾)‘𝑆) ⊆ (((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∩ 𝑌))))
76	75	com23 87	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ 𝑜 ∈ 𝐽) → (((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌) → (𝑜 ⊆ (𝑆 ∪ (𝑋 ∖ 𝑌)) → ((int‘𝐾)‘𝑆) ⊆ (((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∩ 𝑌))))
77	76	impr 460	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌))) → (𝑜 ⊆ (𝑆 ∪ (𝑋 ∖ 𝑌)) → ((int‘𝐾)‘𝑆) ⊆ (((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∩ 𝑌)))
78	59, 77	mpd 16	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌))) → ((int‘𝐾)‘𝑆) ⊆ (((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∩ 𝑌))
79	28, 78	rexlimddv 3170	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((int‘𝐾)‘𝑆) ⊆ (((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∩ 𝑌))
80	1, 11	eqeltrid 2865	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝐾 ∈ Top)
81	8	3adant3 1150	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑌 ∈ V)
82	63, 65, 66	sylanblc 601	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (𝑆 ∪ (𝑋 ∖ 𝑌)) ⊆ 𝑋)
83	4	ntropn 23347	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑆 ∪ (𝑋 ∖ 𝑌)) ⊆ 𝑋) → ((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∈ 𝐽)
84	19, 82, 83	syl2anc 596	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∈ 𝐽)
85		elrestr 17579	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ∈ V ∧ ((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∈ 𝐽) → (((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∩ 𝑌) ∈ (𝐽 ↾t 𝑌))
86	19, 81, 84, 85	syl3anc 1398	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∩ 𝑌) ∈ (𝐽 ↾t 𝑌))
87	86, 1	eleqtrrdi 2872	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∩ 𝑌) ∈ 𝐾)
88	4	ntrss2 23355	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑆 ∪ (𝑋 ∖ 𝑌)) ⊆ 𝑋) → ((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ⊆ (𝑆 ∪ (𝑋 ∖ 𝑌)))
89	19, 82, 88	syl2anc 596	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ⊆ (𝑆 ∪ (𝑋 ∖ 𝑌)))
90	89	ssrind 4189	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∩ 𝑌) ⊆ ((𝑆 ∪ (𝑋 ∖ 𝑌)) ∩ 𝑌))
91		elin 3915	. . . . . . 7 ⊢ (𝑥 ∈ ((𝑆 ∪ (𝑋 ∖ 𝑌)) ∩ 𝑌) ↔ (𝑥 ∈ (𝑆 ∪ (𝑋 ∖ 𝑌)) ∧ 𝑥 ∈ 𝑌))
92		elun 4100	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑆 ∪ (𝑋 ∖ 𝑌)) ↔ (𝑥 ∈ 𝑆 ∨ 𝑥 ∈ (𝑋 ∖ 𝑌)))
93		orcom 884	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑆 ∨ 𝑥 ∈ (𝑋 ∖ 𝑌)) ↔ (𝑥 ∈ (𝑋 ∖ 𝑌) ∨ 𝑥 ∈ 𝑆))
94		df-or 862	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝑋 ∖ 𝑌) ∨ 𝑥 ∈ 𝑆) ↔ (¬ 𝑥 ∈ (𝑋 ∖ 𝑌) → 𝑥 ∈ 𝑆))
95	93, 94	bitri 278	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑆 ∨ 𝑥 ∈ (𝑋 ∖ 𝑌)) ↔ (¬ 𝑥 ∈ (𝑋 ∖ 𝑌) → 𝑥 ∈ 𝑆))
96	92, 95	bitri 278	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑆 ∪ (𝑋 ∖ 𝑌)) ↔ (¬ 𝑥 ∈ (𝑋 ∖ 𝑌) → 𝑥 ∈ 𝑆))
97	96	anbi1i 636	. . . . . . 7 ⊢ ((𝑥 ∈ (𝑆 ∪ (𝑋 ∖ 𝑌)) ∧ 𝑥 ∈ 𝑌) ↔ ((¬ 𝑥 ∈ (𝑋 ∖ 𝑌) → 𝑥 ∈ 𝑆) ∧ 𝑥 ∈ 𝑌))
98	91, 97	bitri 278	. . . . . 6 ⊢ (𝑥 ∈ ((𝑆 ∪ (𝑋 ∖ 𝑌)) ∩ 𝑌) ↔ ((¬ 𝑥 ∈ (𝑋 ∖ 𝑌) → 𝑥 ∈ 𝑆) ∧ 𝑥 ∈ 𝑌))
99		elndif 4080	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑌 → ¬ 𝑥 ∈ (𝑋 ∖ 𝑌))
100		pm2.27 43	. . . . . . . . 9 ⊢ (¬ 𝑥 ∈ (𝑋 ∖ 𝑌) → ((¬ 𝑥 ∈ (𝑋 ∖ 𝑌) → 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑆))
101	99, 100	syl 18	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑌 → ((¬ 𝑥 ∈ (𝑋 ∖ 𝑌) → 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑆))
102	101	impcom 413	. . . . . . 7 ⊢ (((¬ 𝑥 ∈ (𝑋 ∖ 𝑌) → 𝑥 ∈ 𝑆) ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ 𝑆)
103	102	a1i 11	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (((¬ 𝑥 ∈ (𝑋 ∖ 𝑌) → 𝑥 ∈ 𝑆) ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ 𝑆))
104	98, 103	biimtrid 245	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (𝑥 ∈ ((𝑆 ∪ (𝑋 ∖ 𝑌)) ∩ 𝑌) → 𝑥 ∈ 𝑆))
105	104	ssrdv 3937	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((𝑆 ∪ (𝑋 ∖ 𝑌)) ∩ 𝑌) ⊆ 𝑆)
106	90, 105	sstrd 3941	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∩ 𝑌) ⊆ 𝑆)
107	54	ssntr 23356	. . 3 ⊢ (((𝐾 ∈ Top ∧ 𝑆 ⊆ ∪ (𝐽 ↾t 𝑌)) ∧ ((((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∩ 𝑌) ∈ 𝐾 ∧ (((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∩ 𝑌) ⊆ 𝑆)) → (((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∩ 𝑌) ⊆ ((int‘𝐾)‘𝑆))
108	80, 14, 87, 106, 107	syl22anc 852	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∩ 𝑌) ⊆ ((int‘𝐾)‘𝑆))
109	79, 108	eqssd 3948	1 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((int‘𝐾)‘𝑆) = (((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∩ 𝑌))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∃wrex 3087  Vcvv 3451   ∖ cdif 3896   ∪ cun 3897   ∩ cin 3898   ⊆ wss 3899  ∪ cuni 4867  ‘cfv 6531  (class class class)co 7412   ↾_t crest 17571  Topctop 23191  intcnt 23315

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 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  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-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-1st 7990  df-2nd 7991  df-en 8958  df-fin 8961  df-fi 9387  df-rest 17573  df-topgen 17594  df-top 23192  df-topon 23209  df-bases 23244  df-ntr 23318

This theorem is used by:  llycmpkgen2  23849  dvreslem  26209  dvres2lem  26210  dvaddbr  26238  dvmulbr  26239  dvcnvrelem2  26318  limciccioolb  46577  limcicciooub  46591  ioccncflimc  46839  icocncflimc  46843  cncfiooicclem1  46847  fourierdlem62  47122