Theorem restntr 23085

Description: An interior in a subspace topology. Willard in General Topology says that there is no analogue of restcls 23084 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 6900	. . . . . 6 ⊢ (int‘𝐾) = (int‘(𝐽 ↾t 𝑌))
3	2	fveq1i 6898	. . . . 5 ⊢ ((int‘𝐾)‘𝑆) = ((int‘(𝐽 ↾t 𝑌))‘𝑆)
4		restcls.1	. . . . . . . . . 10 ⊢ 𝑋 = ∪ 𝐽
5	4	topopn 22807	. . . . . . . . 9 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
6		ssexg 5323	. . . . . . . . . 10 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑋 ∈ 𝐽) → 𝑌 ∈ V)
7	6	ancoms 458	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐽 ∧ 𝑌 ⊆ 𝑋) → 𝑌 ∈ V)
8	5, 7	sylan 579	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → 𝑌 ∈ V)
9		resttop 23063	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ∈ V) → (𝐽 ↾t 𝑌) ∈ Top)
10	8, 9	syldan 590	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → (𝐽 ↾t 𝑌) ∈ Top)
11	10	3adant3 1130	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (𝐽 ↾t 𝑌) ∈ Top)
12	4	restuni 23065	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → 𝑌 = ∪ (𝐽 ↾t 𝑌))
13	12	sseq2d 4012	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → (𝑆 ⊆ 𝑌 ↔ 𝑆 ⊆ ∪ (𝐽 ↾t 𝑌)))
14	13	biimp3a 1466	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ ∪ (𝐽 ↾t 𝑌))
15		eqid 2728	. . . . . . 7 ⊢ ∪ (𝐽 ↾t 𝑌) = ∪ (𝐽 ↾t 𝑌)
16	15	ntropn 22952	. . . . . 6 ⊢ (((𝐽 ↾t 𝑌) ∈ Top ∧ 𝑆 ⊆ ∪ (𝐽 ↾t 𝑌)) → ((int‘(𝐽 ↾t 𝑌))‘𝑆) ∈ (𝐽 ↾t 𝑌))
17	11, 14, 16	syl2anc 583	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((int‘(𝐽 ↾t 𝑌))‘𝑆) ∈ (𝐽 ↾t 𝑌))
18	3, 17	eqeltrid 2833	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((int‘𝐾)‘𝑆) ∈ (𝐽 ↾t 𝑌))
19		simp1 1134	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝐽 ∈ Top)
20		uniexg 7745	. . . . . . . . 9 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ V)
21	4, 20	eqeltrid 2833	. . . . . . . 8 ⊢ (𝐽 ∈ Top → 𝑋 ∈ V)
22		ssexg 5323	. . . . . . . 8 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑋 ∈ V) → 𝑌 ∈ V)
23	21, 22	sylan2 592	. . . . . . 7 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝐽 ∈ Top) → 𝑌 ∈ V)
24	23	ancoms 458	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → 𝑌 ∈ V)
25	24	3adant3 1130	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑌 ∈ V)
26		elrest 17408	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ∈ V) → (((int‘𝐾)‘𝑆) ∈ (𝐽 ↾t 𝑌) ↔ ∃𝑜 ∈ 𝐽 ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌)))
27	19, 25, 26	syl2anc 583	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (((int‘𝐾)‘𝑆) ∈ (𝐽 ↾t 𝑌) ↔ ∃𝑜 ∈ 𝐽 ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌)))
28	18, 27	mpbid 231	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ∃𝑜 ∈ 𝐽 ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌))
29	4	eltopss 22808	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → 𝑜 ⊆ 𝑋)
30	29	sseld 3979	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → (𝑥 ∈ 𝑜 → 𝑥 ∈ 𝑋))
31	30	adantrr 716	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ (𝑜 ∈ 𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌))) → (𝑥 ∈ 𝑜 → 𝑥 ∈ 𝑋))
32	31	3ad2antl1 1183	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌))) → (𝑥 ∈ 𝑜 → 𝑥 ∈ 𝑋))
33		eldif 3957	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑋 ∖ 𝑌) ↔ (𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ 𝑌))
34	33	simplbi2 500	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋 → (¬ 𝑥 ∈ 𝑌 → 𝑥 ∈ (𝑋 ∖ 𝑌)))
35	34	orrd 862	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑋 → (𝑥 ∈ 𝑌 ∨ 𝑥 ∈ (𝑋 ∖ 𝑌)))
36	32, 35	syl6 35	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌))) → (𝑥 ∈ 𝑜 → (𝑥 ∈ 𝑌 ∨ 𝑥 ∈ (𝑋 ∖ 𝑌))))
37		elin 3963	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝑜 ∩ 𝑌) ↔ (𝑥 ∈ 𝑜 ∧ 𝑥 ∈ 𝑌))
38		eleq2 2818	. . . . . . . . . . . . 13 ⊢ (((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌) → (𝑥 ∈ ((int‘𝐾)‘𝑆) ↔ 𝑥 ∈ (𝑜 ∩ 𝑌)))
39		elun1 4176	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ((int‘𝐾)‘𝑆) → 𝑥 ∈ (((int‘𝐾)‘𝑆) ∪ (𝑋 ∖ 𝑌)))
40	38, 39	syl6bir 254	. . . . . . . . . . . 12 ⊢ (((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌) → (𝑥 ∈ (𝑜 ∩ 𝑌) → 𝑥 ∈ (((int‘𝐾)‘𝑆) ∪ (𝑋 ∖ 𝑌))))
41	40	ad2antll 728	. . . . . . . . . . 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 242	. . . . . . . . . 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 452	. . . . . . . . 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 4177	. . . . . . . . . 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 858	. . . . . . . 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 412	. . . . . . 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 43	. . . . . 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 3986	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌))) → 𝑜 ⊆ (((int‘𝐾)‘𝑆) ∪ (𝑋 ∖ 𝑌)))
50	11	adantr 480	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌))) → (𝐽 ↾t 𝑌) ∈ Top)
51	1, 50	eqeltrid 2833	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌))) → 𝐾 ∈ Top)
52	14	adantr 480	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌))) → 𝑆 ⊆ ∪ (𝐽 ↾t 𝑌))
53	1	unieqi 4920	. . . . . . . . 9 ⊢ ∪ 𝐾 = ∪ (𝐽 ↾t 𝑌)
54	53	eqcomi 2737	. . . . . . . 8 ⊢ ∪ (𝐽 ↾t 𝑌) = ∪ 𝐾
55	54	ntrss2 22960	. . . . . . 7 ⊢ ((𝐾 ∈ Top ∧ 𝑆 ⊆ ∪ (𝐽 ↾t 𝑌)) → ((int‘𝐾)‘𝑆) ⊆ 𝑆)
56	51, 52, 55	syl2anc 583	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌))) → ((int‘𝐾)‘𝑆) ⊆ 𝑆)
57		unss1 4179	. . . . . 6 ⊢ (((int‘𝐾)‘𝑆) ⊆ 𝑆 → (((int‘𝐾)‘𝑆) ∪ (𝑋 ∖ 𝑌)) ⊆ (𝑆 ∪ (𝑋 ∖ 𝑌)))
58	56, 57	syl 17	. . . . 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 3990	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌))) → 𝑜 ⊆ (𝑆 ∪ (𝑋 ∖ 𝑌)))
60		simpl1 1189	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ 𝑜 ⊆ (𝑆 ∪ (𝑋 ∖ 𝑌)))) → 𝐽 ∈ Top)
61		sstr 3988	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝑋) → 𝑆 ⊆ 𝑋)
62	61	ancoms 458	. . . . . . . . . . . . 13 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ 𝑋)
63	62	3adant1 1128	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ 𝑋)
64	63	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ 𝑜 ⊆ (𝑆 ∪ (𝑋 ∖ 𝑌)))) → 𝑆 ⊆ 𝑋)
65		difss 4130	. . . . . . . . . . 11 ⊢ (𝑋 ∖ 𝑌) ⊆ 𝑋
66		unss 4184	. . . . . . . . . . 11 ⊢ ((𝑆 ⊆ 𝑋 ∧ (𝑋 ∖ 𝑌) ⊆ 𝑋) ↔ (𝑆 ∪ (𝑋 ∖ 𝑌)) ⊆ 𝑋)
67	64, 65, 66	sylanblc 588	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ 𝑜 ⊆ (𝑆 ∪ (𝑋 ∖ 𝑌)))) → (𝑆 ∪ (𝑋 ∖ 𝑌)) ⊆ 𝑋)
68		simprl 770	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ 𝑜 ⊆ (𝑆 ∪ (𝑋 ∖ 𝑌)))) → 𝑜 ∈ 𝐽)
69		simprr 772	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ 𝑜 ⊆ (𝑆 ∪ (𝑋 ∖ 𝑌)))) → 𝑜 ⊆ (𝑆 ∪ (𝑋 ∖ 𝑌)))
70	4	ssntr 22961	. . . . . . . . . 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 838	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ 𝑜 ⊆ (𝑆 ∪ (𝑋 ∖ 𝑌)))) → 𝑜 ⊆ ((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))))
72	71	ssrind 4236	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ 𝑜 ⊆ (𝑆 ∪ (𝑋 ∖ 𝑌)))) → (𝑜 ∩ 𝑌) ⊆ (((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∩ 𝑌))
73		sseq1 4005	. . . . . . . 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 246	. . . . . . 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 456	. . . . . 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 86	. . . . 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 454	. . . 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 15	. . 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 3158	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((int‘𝐾)‘𝑆) ⊆ (((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∩ 𝑌))
80	1, 11	eqeltrid 2833	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝐾 ∈ Top)
81	8	3adant3 1130	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑌 ∈ V)
82	63, 65, 66	sylanblc 588	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (𝑆 ∪ (𝑋 ∖ 𝑌)) ⊆ 𝑋)
83	4	ntropn 22952	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑆 ∪ (𝑋 ∖ 𝑌)) ⊆ 𝑋) → ((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∈ 𝐽)
84	19, 82, 83	syl2anc 583	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∈ 𝐽)
85		elrestr 17409	. . . . 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 1369	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∩ 𝑌) ∈ (𝐽 ↾t 𝑌))
87	86, 1	eleqtrrdi 2840	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∩ 𝑌) ∈ 𝐾)
88	4	ntrss2 22960	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑆 ∪ (𝑋 ∖ 𝑌)) ⊆ 𝑋) → ((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ⊆ (𝑆 ∪ (𝑋 ∖ 𝑌)))
89	19, 82, 88	syl2anc 583	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ⊆ (𝑆 ∪ (𝑋 ∖ 𝑌)))
90	89	ssrind 4236	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∩ 𝑌) ⊆ ((𝑆 ∪ (𝑋 ∖ 𝑌)) ∩ 𝑌))
91		elin 3963	. . . . . . 7 ⊢ (𝑥 ∈ ((𝑆 ∪ (𝑋 ∖ 𝑌)) ∩ 𝑌) ↔ (𝑥 ∈ (𝑆 ∪ (𝑋 ∖ 𝑌)) ∧ 𝑥 ∈ 𝑌))
92		elun 4147	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑆 ∪ (𝑋 ∖ 𝑌)) ↔ (𝑥 ∈ 𝑆 ∨ 𝑥 ∈ (𝑋 ∖ 𝑌)))
93		orcom 869	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑆 ∨ 𝑥 ∈ (𝑋 ∖ 𝑌)) ↔ (𝑥 ∈ (𝑋 ∖ 𝑌) ∨ 𝑥 ∈ 𝑆))
94		df-or 847	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝑋 ∖ 𝑌) ∨ 𝑥 ∈ 𝑆) ↔ (¬ 𝑥 ∈ (𝑋 ∖ 𝑌) → 𝑥 ∈ 𝑆))
95	93, 94	bitri 275	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑆 ∨ 𝑥 ∈ (𝑋 ∖ 𝑌)) ↔ (¬ 𝑥 ∈ (𝑋 ∖ 𝑌) → 𝑥 ∈ 𝑆))
96	92, 95	bitri 275	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑆 ∪ (𝑋 ∖ 𝑌)) ↔ (¬ 𝑥 ∈ (𝑋 ∖ 𝑌) → 𝑥 ∈ 𝑆))
97	96	anbi1i 623	. . . . . . 7 ⊢ ((𝑥 ∈ (𝑆 ∪ (𝑋 ∖ 𝑌)) ∧ 𝑥 ∈ 𝑌) ↔ ((¬ 𝑥 ∈ (𝑋 ∖ 𝑌) → 𝑥 ∈ 𝑆) ∧ 𝑥 ∈ 𝑌))
98	91, 97	bitri 275	. . . . . 6 ⊢ (𝑥 ∈ ((𝑆 ∪ (𝑋 ∖ 𝑌)) ∩ 𝑌) ↔ ((¬ 𝑥 ∈ (𝑋 ∖ 𝑌) → 𝑥 ∈ 𝑆) ∧ 𝑥 ∈ 𝑌))
99		elndif 4127	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑌 → ¬ 𝑥 ∈ (𝑋 ∖ 𝑌))
100		pm2.27 42	. . . . . . . . 9 ⊢ (¬ 𝑥 ∈ (𝑋 ∖ 𝑌) → ((¬ 𝑥 ∈ (𝑋 ∖ 𝑌) → 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑆))
101	99, 100	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑌 → ((¬ 𝑥 ∈ (𝑋 ∖ 𝑌) → 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑆))
102	101	impcom 407	. . . . . . 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 241	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (𝑥 ∈ ((𝑆 ∪ (𝑋 ∖ 𝑌)) ∩ 𝑌) → 𝑥 ∈ 𝑆))
105	104	ssrdv 3986	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((𝑆 ∪ (𝑋 ∖ 𝑌)) ∩ 𝑌) ⊆ 𝑆)
106	90, 105	sstrd 3990	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∩ 𝑌) ⊆ 𝑆)
107	54	ssntr 22961	. . 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 838	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∩ 𝑌) ⊆ ((int‘𝐾)‘𝑆))
109	79, 108	eqssd 3997	1 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((int‘𝐾)‘𝑆) = (((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∩ 𝑌))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 846   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  ∃wrex 3067  Vcvv 3471   ∖ cdif 3944   ∪ cun 3945   ∩ cin 3946   ⊆ wss 3947  ∪ cuni 4908  ‘cfv 6548  (class class class)co 7420   ↾_t crest 17401  Topctop 22794  intcnt 22920

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-1st 7993  df-2nd 7994  df-en 8964  df-fin 8967  df-fi 9434  df-rest 17403  df-topgen 17424  df-top 22795  df-topon 22812  df-bases 22848  df-ntr 22923

This theorem is referenced by:  llycmpkgen2  23453  dvreslem  25837  dvres2lem  25838  dvaddbr  25867  dvmulbr  25868  dvmulbrOLD  25869  dvcnvrelem2  25950  limciccioolb  45009  limcicciooub  45025  ioccncflimc  45273  icocncflimc  45277  cncfiooicclem1  45281  fourierdlem62  45556