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Theorem restntr 21208
Description: An interior in a subspace topology. Willard in General Topology says that there is no analogue of restcls 21207 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 = (𝐽t 𝑌)
Assertion
Ref Expression
restntr ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((int‘𝐾)‘𝑆) = (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌))

Proof of Theorem restntr
Dummy variables 𝑥 𝑜 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 restcls.2 . . . . . . 7 𝐾 = (𝐽t 𝑌)
21fveq2i 6418 . . . . . 6 (int‘𝐾) = (int‘(𝐽t 𝑌))
32fveq1i 6416 . . . . 5 ((int‘𝐾)‘𝑆) = ((int‘(𝐽t 𝑌))‘𝑆)
4 restcls.1 . . . . . . . . . 10 𝑋 = 𝐽
54topopn 20932 . . . . . . . . 9 (𝐽 ∈ Top → 𝑋𝐽)
6 ssexg 5010 . . . . . . . . . 10 ((𝑌𝑋𝑋𝐽) → 𝑌 ∈ V)
76ancoms 448 . . . . . . . . 9 ((𝑋𝐽𝑌𝑋) → 𝑌 ∈ V)
85, 7sylan 571 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑌𝑋) → 𝑌 ∈ V)
9 resttop 21186 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑌 ∈ V) → (𝐽t 𝑌) ∈ Top)
108, 9syldan 581 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑌𝑋) → (𝐽t 𝑌) ∈ Top)
11103adant3 1155 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (𝐽t 𝑌) ∈ Top)
124restuni 21188 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑌𝑋) → 𝑌 = (𝐽t 𝑌))
1312sseq2d 3841 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑌𝑋) → (𝑆𝑌𝑆 (𝐽t 𝑌)))
1413biimp3a 1586 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑆 (𝐽t 𝑌))
15 eqid 2817 . . . . . . 7 (𝐽t 𝑌) = (𝐽t 𝑌)
1615ntropn 21075 . . . . . 6 (((𝐽t 𝑌) ∈ Top ∧ 𝑆 (𝐽t 𝑌)) → ((int‘(𝐽t 𝑌))‘𝑆) ∈ (𝐽t 𝑌))
1711, 14, 16syl2anc 575 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((int‘(𝐽t 𝑌))‘𝑆) ∈ (𝐽t 𝑌))
183, 17syl5eqel 2900 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((int‘𝐾)‘𝑆) ∈ (𝐽t 𝑌))
19 simp1 1159 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝐽 ∈ Top)
20 uniexg 7192 . . . . . . . . 9 (𝐽 ∈ Top → 𝐽 ∈ V)
214, 20syl5eqel 2900 . . . . . . . 8 (𝐽 ∈ Top → 𝑋 ∈ V)
22 ssexg 5010 . . . . . . . 8 ((𝑌𝑋𝑋 ∈ V) → 𝑌 ∈ V)
2321, 22sylan2 582 . . . . . . 7 ((𝑌𝑋𝐽 ∈ Top) → 𝑌 ∈ V)
2423ancoms 448 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋) → 𝑌 ∈ V)
25243adant3 1155 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑌 ∈ V)
26 elrest 16300 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌 ∈ V) → (((int‘𝐾)‘𝑆) ∈ (𝐽t 𝑌) ↔ ∃𝑜𝐽 ((int‘𝐾)‘𝑆) = (𝑜𝑌)))
2719, 25, 26syl2anc 575 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (((int‘𝐾)‘𝑆) ∈ (𝐽t 𝑌) ↔ ∃𝑜𝐽 ((int‘𝐾)‘𝑆) = (𝑜𝑌)))
2818, 27mpbid 223 . . 3 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ∃𝑜𝐽 ((int‘𝐾)‘𝑆) = (𝑜𝑌))
294eltopss 20933 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑜𝐽) → 𝑜𝑋)
3029sseld 3808 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑜𝐽) → (𝑥𝑜𝑥𝑋))
3130adantrr 699 . . . . . . . . 9 ((𝐽 ∈ Top ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) → (𝑥𝑜𝑥𝑋))
32313ad2antl1 1229 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) → (𝑥𝑜𝑥𝑋))
33 eldif 3790 . . . . . . . . . 10 (𝑥 ∈ (𝑋𝑌) ↔ (𝑥𝑋 ∧ ¬ 𝑥𝑌))
3433simplbi2 490 . . . . . . . . 9 (𝑥𝑋 → (¬ 𝑥𝑌𝑥 ∈ (𝑋𝑌)))
3534orrd 881 . . . . . . . 8 (𝑥𝑋 → (𝑥𝑌𝑥 ∈ (𝑋𝑌)))
3632, 35syl6 35 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) → (𝑥𝑜 → (𝑥𝑌𝑥 ∈ (𝑋𝑌))))
37 elin 4006 . . . . . . . . . . 11 (𝑥 ∈ (𝑜𝑌) ↔ (𝑥𝑜𝑥𝑌))
38 eleq2 2885 . . . . . . . . . . . . 13 (((int‘𝐾)‘𝑆) = (𝑜𝑌) → (𝑥 ∈ ((int‘𝐾)‘𝑆) ↔ 𝑥 ∈ (𝑜𝑌)))
39 elun1 3990 . . . . . . . . . . . . 13 (𝑥 ∈ ((int‘𝐾)‘𝑆) → 𝑥 ∈ (((int‘𝐾)‘𝑆) ∪ (𝑋𝑌)))
4038, 39syl6bir 245 . . . . . . . . . . . 12 (((int‘𝐾)‘𝑆) = (𝑜𝑌) → (𝑥 ∈ (𝑜𝑌) → 𝑥 ∈ (((int‘𝐾)‘𝑆) ∪ (𝑋𝑌))))
4140ad2antll 711 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) → (𝑥 ∈ (𝑜𝑌) → 𝑥 ∈ (((int‘𝐾)‘𝑆) ∪ (𝑋𝑌))))
4237, 41syl5bir 234 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) → ((𝑥𝑜𝑥𝑌) → 𝑥 ∈ (((int‘𝐾)‘𝑆) ∪ (𝑋𝑌))))
4342expdimp 442 . . . . . . . . 9 ((((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) ∧ 𝑥𝑜) → (𝑥𝑌𝑥 ∈ (((int‘𝐾)‘𝑆) ∪ (𝑋𝑌))))
44 elun2 3991 . . . . . . . . . 10 (𝑥 ∈ (𝑋𝑌) → 𝑥 ∈ (((int‘𝐾)‘𝑆) ∪ (𝑋𝑌)))
4544a1i 11 . . . . . . . . 9 ((((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) ∧ 𝑥𝑜) → (𝑥 ∈ (𝑋𝑌) → 𝑥 ∈ (((int‘𝐾)‘𝑆) ∪ (𝑋𝑌))))
4643, 45jaod 877 . . . . . . . 8 ((((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) ∧ 𝑥𝑜) → ((𝑥𝑌𝑥 ∈ (𝑋𝑌)) → 𝑥 ∈ (((int‘𝐾)‘𝑆) ∪ (𝑋𝑌))))
4746ex 399 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) → (𝑥𝑜 → ((𝑥𝑌𝑥 ∈ (𝑋𝑌)) → 𝑥 ∈ (((int‘𝐾)‘𝑆) ∪ (𝑋𝑌)))))
4836, 47mpdd 43 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) → (𝑥𝑜𝑥 ∈ (((int‘𝐾)‘𝑆) ∪ (𝑋𝑌))))
4948ssrdv 3815 . . . . 5 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) → 𝑜 ⊆ (((int‘𝐾)‘𝑆) ∪ (𝑋𝑌)))
5011adantr 468 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) → (𝐽t 𝑌) ∈ Top)
511, 50syl5eqel 2900 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) → 𝐾 ∈ Top)
5214adantr 468 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) → 𝑆 (𝐽t 𝑌))
531unieqi 4650 . . . . . . . . 9 𝐾 = (𝐽t 𝑌)
5453eqcomi 2826 . . . . . . . 8 (𝐽t 𝑌) = 𝐾
5554ntrss2 21083 . . . . . . 7 ((𝐾 ∈ Top ∧ 𝑆 (𝐽t 𝑌)) → ((int‘𝐾)‘𝑆) ⊆ 𝑆)
5651, 52, 55syl2anc 575 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) → ((int‘𝐾)‘𝑆) ⊆ 𝑆)
57 unss1 3992 . . . . . 6 (((int‘𝐾)‘𝑆) ⊆ 𝑆 → (((int‘𝐾)‘𝑆) ∪ (𝑋𝑌)) ⊆ (𝑆 ∪ (𝑋𝑌)))
5856, 57syl 17 . . . . 5 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) → (((int‘𝐾)‘𝑆) ∪ (𝑋𝑌)) ⊆ (𝑆 ∪ (𝑋𝑌)))
5949, 58sstrd 3819 . . . 4 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) → 𝑜 ⊆ (𝑆 ∪ (𝑋𝑌)))
60 simpl1 1235 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽𝑜 ⊆ (𝑆 ∪ (𝑋𝑌)))) → 𝐽 ∈ Top)
61 sstr 3817 . . . . . . . . . . . . . 14 ((𝑆𝑌𝑌𝑋) → 𝑆𝑋)
6261ancoms 448 . . . . . . . . . . . . 13 ((𝑌𝑋𝑆𝑌) → 𝑆𝑋)
63623adant1 1153 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑆𝑋)
6463adantr 468 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽𝑜 ⊆ (𝑆 ∪ (𝑋𝑌)))) → 𝑆𝑋)
65 difss 3947 . . . . . . . . . . 11 (𝑋𝑌) ⊆ 𝑋
66 unss 3997 . . . . . . . . . . 11 ((𝑆𝑋 ∧ (𝑋𝑌) ⊆ 𝑋) ↔ (𝑆 ∪ (𝑋𝑌)) ⊆ 𝑋)
6764, 65, 66sylanblc 579 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽𝑜 ⊆ (𝑆 ∪ (𝑋𝑌)))) → (𝑆 ∪ (𝑋𝑌)) ⊆ 𝑋)
68 simprl 778 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽𝑜 ⊆ (𝑆 ∪ (𝑋𝑌)))) → 𝑜𝐽)
69 simprr 780 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽𝑜 ⊆ (𝑆 ∪ (𝑋𝑌)))) → 𝑜 ⊆ (𝑆 ∪ (𝑋𝑌)))
704ssntr 21084 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ (𝑆 ∪ (𝑋𝑌)) ⊆ 𝑋) ∧ (𝑜𝐽𝑜 ⊆ (𝑆 ∪ (𝑋𝑌)))) → 𝑜 ⊆ ((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))))
7160, 67, 68, 69, 70syl22anc 858 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽𝑜 ⊆ (𝑆 ∪ (𝑋𝑌)))) → 𝑜 ⊆ ((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))))
7271ssrind 4047 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽𝑜 ⊆ (𝑆 ∪ (𝑋𝑌)))) → (𝑜𝑌) ⊆ (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌))
73 sseq1 3834 . . . . . . . 8 (((int‘𝐾)‘𝑆) = (𝑜𝑌) → (((int‘𝐾)‘𝑆) ⊆ (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌) ↔ (𝑜𝑌) ⊆ (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌)))
7472, 73syl5ibrcom 238 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽𝑜 ⊆ (𝑆 ∪ (𝑋𝑌)))) → (((int‘𝐾)‘𝑆) = (𝑜𝑌) → ((int‘𝐾)‘𝑆) ⊆ (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌)))
7574expr 446 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ 𝑜𝐽) → (𝑜 ⊆ (𝑆 ∪ (𝑋𝑌)) → (((int‘𝐾)‘𝑆) = (𝑜𝑌) → ((int‘𝐾)‘𝑆) ⊆ (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌))))
7675com23 86 . . . . 5 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ 𝑜𝐽) → (((int‘𝐾)‘𝑆) = (𝑜𝑌) → (𝑜 ⊆ (𝑆 ∪ (𝑋𝑌)) → ((int‘𝐾)‘𝑆) ⊆ (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌))))
7776impr 444 . . . 4 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) → (𝑜 ⊆ (𝑆 ∪ (𝑋𝑌)) → ((int‘𝐾)‘𝑆) ⊆ (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌)))
7859, 77mpd 15 . . 3 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) → ((int‘𝐾)‘𝑆) ⊆ (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌))
7928, 78rexlimddv 3234 . 2 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((int‘𝐾)‘𝑆) ⊆ (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌))
801, 11syl5eqel 2900 . . 3 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝐾 ∈ Top)
8183adant3 1155 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑌 ∈ V)
8263, 65, 66sylanblc 579 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (𝑆 ∪ (𝑋𝑌)) ⊆ 𝑋)
834ntropn 21075 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑆 ∪ (𝑋𝑌)) ⊆ 𝑋) → ((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∈ 𝐽)
8419, 82, 83syl2anc 575 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∈ 𝐽)
85 elrestr 16301 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌 ∈ V ∧ ((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∈ 𝐽) → (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌) ∈ (𝐽t 𝑌))
8619, 81, 84, 85syl3anc 1483 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌) ∈ (𝐽t 𝑌))
8786, 1syl6eleqr 2907 . . 3 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌) ∈ 𝐾)
884ntrss2 21083 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑆 ∪ (𝑋𝑌)) ⊆ 𝑋) → ((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ⊆ (𝑆 ∪ (𝑋𝑌)))
8919, 82, 88syl2anc 575 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ⊆ (𝑆 ∪ (𝑋𝑌)))
9089ssrind 4047 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌) ⊆ ((𝑆 ∪ (𝑋𝑌)) ∩ 𝑌))
91 elin 4006 . . . . . . 7 (𝑥 ∈ ((𝑆 ∪ (𝑋𝑌)) ∩ 𝑌) ↔ (𝑥 ∈ (𝑆 ∪ (𝑋𝑌)) ∧ 𝑥𝑌))
92 elun 3963 . . . . . . . . 9 (𝑥 ∈ (𝑆 ∪ (𝑋𝑌)) ↔ (𝑥𝑆𝑥 ∈ (𝑋𝑌)))
93 orcom 888 . . . . . . . . . 10 ((𝑥𝑆𝑥 ∈ (𝑋𝑌)) ↔ (𝑥 ∈ (𝑋𝑌) ∨ 𝑥𝑆))
94 df-or 866 . . . . . . . . . 10 ((𝑥 ∈ (𝑋𝑌) ∨ 𝑥𝑆) ↔ (¬ 𝑥 ∈ (𝑋𝑌) → 𝑥𝑆))
9593, 94bitri 266 . . . . . . . . 9 ((𝑥𝑆𝑥 ∈ (𝑋𝑌)) ↔ (¬ 𝑥 ∈ (𝑋𝑌) → 𝑥𝑆))
9692, 95bitri 266 . . . . . . . 8 (𝑥 ∈ (𝑆 ∪ (𝑋𝑌)) ↔ (¬ 𝑥 ∈ (𝑋𝑌) → 𝑥𝑆))
9796anbi1i 612 . . . . . . 7 ((𝑥 ∈ (𝑆 ∪ (𝑋𝑌)) ∧ 𝑥𝑌) ↔ ((¬ 𝑥 ∈ (𝑋𝑌) → 𝑥𝑆) ∧ 𝑥𝑌))
9891, 97bitri 266 . . . . . 6 (𝑥 ∈ ((𝑆 ∪ (𝑋𝑌)) ∩ 𝑌) ↔ ((¬ 𝑥 ∈ (𝑋𝑌) → 𝑥𝑆) ∧ 𝑥𝑌))
99 elndif 3944 . . . . . . . . 9 (𝑥𝑌 → ¬ 𝑥 ∈ (𝑋𝑌))
100 pm2.27 42 . . . . . . . . 9 𝑥 ∈ (𝑋𝑌) → ((¬ 𝑥 ∈ (𝑋𝑌) → 𝑥𝑆) → 𝑥𝑆))
10199, 100syl 17 . . . . . . . 8 (𝑥𝑌 → ((¬ 𝑥 ∈ (𝑋𝑌) → 𝑥𝑆) → 𝑥𝑆))
102101impcom 396 . . . . . . 7 (((¬ 𝑥 ∈ (𝑋𝑌) → 𝑥𝑆) ∧ 𝑥𝑌) → 𝑥𝑆)
103102a1i 11 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (((¬ 𝑥 ∈ (𝑋𝑌) → 𝑥𝑆) ∧ 𝑥𝑌) → 𝑥𝑆))
10498, 103syl5bi 233 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (𝑥 ∈ ((𝑆 ∪ (𝑋𝑌)) ∩ 𝑌) → 𝑥𝑆))
105104ssrdv 3815 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((𝑆 ∪ (𝑋𝑌)) ∩ 𝑌) ⊆ 𝑆)
10690, 105sstrd 3819 . . 3 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌) ⊆ 𝑆)
10754ssntr 21084 . . 3 (((𝐾 ∈ Top ∧ 𝑆 (𝐽t 𝑌)) ∧ ((((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌) ∈ 𝐾 ∧ (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌) ⊆ 𝑆)) → (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌) ⊆ ((int‘𝐾)‘𝑆))
10880, 14, 87, 106, 107syl22anc 858 . 2 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌) ⊆ ((int‘𝐾)‘𝑆))
10979, 108eqssd 3826 1 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((int‘𝐾)‘𝑆) = (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 865  w3a 1100   = wceq 1637  wcel 2157  wrex 3108  Vcvv 3402  cdif 3777  cun 3778  cin 3779  wss 3780   cuni 4641  cfv 6108  (class class class)co 6881  t crest 16293  Topctop 20919  intcnt 21043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-rep 4975  ax-sep 4986  ax-nul 4994  ax-pow 5046  ax-pr 5107  ax-un 7186
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-pss 3796  df-nul 4128  df-if 4291  df-pw 4364  df-sn 4382  df-pr 4384  df-tp 4386  df-op 4388  df-uni 4642  df-int 4681  df-iun 4725  df-br 4856  df-opab 4918  df-mpt 4935  df-tr 4958  df-id 5230  df-eprel 5235  df-po 5243  df-so 5244  df-fr 5281  df-we 5283  df-xp 5328  df-rel 5329  df-cnv 5330  df-co 5331  df-dm 5332  df-rn 5333  df-res 5334  df-ima 5335  df-pred 5904  df-ord 5950  df-on 5951  df-lim 5952  df-suc 5953  df-iota 6071  df-fun 6110  df-fn 6111  df-f 6112  df-f1 6113  df-fo 6114  df-f1o 6115  df-fv 6116  df-ov 6884  df-oprab 6885  df-mpt2 6886  df-om 7303  df-1st 7405  df-2nd 7406  df-wrecs 7649  df-recs 7711  df-rdg 7749  df-oadd 7807  df-er 7986  df-en 8200  df-fin 8203  df-fi 8563  df-rest 16295  df-topgen 16316  df-top 20920  df-topon 20937  df-bases 20972  df-ntr 21046
This theorem is referenced by:  llycmpkgen2  21575  dvreslem  23897  dvres2lem  23898  dvaddbr  23925  dvmulbr  23926  dvcnvrelem2  24005  limciccioolb  40338  limcicciooub  40354  ioccncflimc  40583  icocncflimc  40587  cncfiooicclem1  40591  fourierdlem62  40869
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