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Theorem restntr 22333
Description: An interior in a subspace topology. Willard in General Topology says that there is no analogue of restcls 22332 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 6777 . . . . . 6 (int‘𝐾) = (int‘(𝐽t 𝑌))
32fveq1i 6775 . . . . 5 ((int‘𝐾)‘𝑆) = ((int‘(𝐽t 𝑌))‘𝑆)
4 restcls.1 . . . . . . . . . 10 𝑋 = 𝐽
54topopn 22055 . . . . . . . . 9 (𝐽 ∈ Top → 𝑋𝐽)
6 ssexg 5247 . . . . . . . . . 10 ((𝑌𝑋𝑋𝐽) → 𝑌 ∈ V)
76ancoms 459 . . . . . . . . 9 ((𝑋𝐽𝑌𝑋) → 𝑌 ∈ V)
85, 7sylan 580 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑌𝑋) → 𝑌 ∈ V)
9 resttop 22311 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑌 ∈ V) → (𝐽t 𝑌) ∈ Top)
108, 9syldan 591 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑌𝑋) → (𝐽t 𝑌) ∈ Top)
11103adant3 1131 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (𝐽t 𝑌) ∈ Top)
124restuni 22313 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑌𝑋) → 𝑌 = (𝐽t 𝑌))
1312sseq2d 3953 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑌𝑋) → (𝑆𝑌𝑆 (𝐽t 𝑌)))
1413biimp3a 1468 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑆 (𝐽t 𝑌))
15 eqid 2738 . . . . . . 7 (𝐽t 𝑌) = (𝐽t 𝑌)
1615ntropn 22200 . . . . . 6 (((𝐽t 𝑌) ∈ Top ∧ 𝑆 (𝐽t 𝑌)) → ((int‘(𝐽t 𝑌))‘𝑆) ∈ (𝐽t 𝑌))
1711, 14, 16syl2anc 584 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((int‘(𝐽t 𝑌))‘𝑆) ∈ (𝐽t 𝑌))
183, 17eqeltrid 2843 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((int‘𝐾)‘𝑆) ∈ (𝐽t 𝑌))
19 simp1 1135 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝐽 ∈ Top)
20 uniexg 7593 . . . . . . . . 9 (𝐽 ∈ Top → 𝐽 ∈ V)
214, 20eqeltrid 2843 . . . . . . . 8 (𝐽 ∈ Top → 𝑋 ∈ V)
22 ssexg 5247 . . . . . . . 8 ((𝑌𝑋𝑋 ∈ V) → 𝑌 ∈ V)
2321, 22sylan2 593 . . . . . . 7 ((𝑌𝑋𝐽 ∈ Top) → 𝑌 ∈ V)
2423ancoms 459 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋) → 𝑌 ∈ V)
25243adant3 1131 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑌 ∈ V)
26 elrest 17138 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌 ∈ V) → (((int‘𝐾)‘𝑆) ∈ (𝐽t 𝑌) ↔ ∃𝑜𝐽 ((int‘𝐾)‘𝑆) = (𝑜𝑌)))
2719, 25, 26syl2anc 584 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (((int‘𝐾)‘𝑆) ∈ (𝐽t 𝑌) ↔ ∃𝑜𝐽 ((int‘𝐾)‘𝑆) = (𝑜𝑌)))
2818, 27mpbid 231 . . 3 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ∃𝑜𝐽 ((int‘𝐾)‘𝑆) = (𝑜𝑌))
294eltopss 22056 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑜𝐽) → 𝑜𝑋)
3029sseld 3920 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑜𝐽) → (𝑥𝑜𝑥𝑋))
3130adantrr 714 . . . . . . . . 9 ((𝐽 ∈ Top ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) → (𝑥𝑜𝑥𝑋))
32313ad2antl1 1184 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) → (𝑥𝑜𝑥𝑋))
33 eldif 3897 . . . . . . . . . 10 (𝑥 ∈ (𝑋𝑌) ↔ (𝑥𝑋 ∧ ¬ 𝑥𝑌))
3433simplbi2 501 . . . . . . . . 9 (𝑥𝑋 → (¬ 𝑥𝑌𝑥 ∈ (𝑋𝑌)))
3534orrd 860 . . . . . . . 8 (𝑥𝑋 → (𝑥𝑌𝑥 ∈ (𝑋𝑌)))
3632, 35syl6 35 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) → (𝑥𝑜 → (𝑥𝑌𝑥 ∈ (𝑋𝑌))))
37 elin 3903 . . . . . . . . . . 11 (𝑥 ∈ (𝑜𝑌) ↔ (𝑥𝑜𝑥𝑌))
38 eleq2 2827 . . . . . . . . . . . . 13 (((int‘𝐾)‘𝑆) = (𝑜𝑌) → (𝑥 ∈ ((int‘𝐾)‘𝑆) ↔ 𝑥 ∈ (𝑜𝑌)))
39 elun1 4110 . . . . . . . . . . . . 13 (𝑥 ∈ ((int‘𝐾)‘𝑆) → 𝑥 ∈ (((int‘𝐾)‘𝑆) ∪ (𝑋𝑌)))
4038, 39syl6bir 253 . . . . . . . . . . . 12 (((int‘𝐾)‘𝑆) = (𝑜𝑌) → (𝑥 ∈ (𝑜𝑌) → 𝑥 ∈ (((int‘𝐾)‘𝑆) ∪ (𝑋𝑌))))
4140ad2antll 726 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) → (𝑥 ∈ (𝑜𝑌) → 𝑥 ∈ (((int‘𝐾)‘𝑆) ∪ (𝑋𝑌))))
4237, 41syl5bir 242 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) → ((𝑥𝑜𝑥𝑌) → 𝑥 ∈ (((int‘𝐾)‘𝑆) ∪ (𝑋𝑌))))
4342expdimp 453 . . . . . . . . 9 ((((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) ∧ 𝑥𝑜) → (𝑥𝑌𝑥 ∈ (((int‘𝐾)‘𝑆) ∪ (𝑋𝑌))))
44 elun2 4111 . . . . . . . . . 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 856 . . . . . . . 8 ((((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) ∧ 𝑥𝑜) → ((𝑥𝑌𝑥 ∈ (𝑋𝑌)) → 𝑥 ∈ (((int‘𝐾)‘𝑆) ∪ (𝑋𝑌))))
4746ex 413 . . . . . . 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 3927 . . . . 5 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) → 𝑜 ⊆ (((int‘𝐾)‘𝑆) ∪ (𝑋𝑌)))
5011adantr 481 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) → (𝐽t 𝑌) ∈ Top)
511, 50eqeltrid 2843 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) → 𝐾 ∈ Top)
5214adantr 481 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) → 𝑆 (𝐽t 𝑌))
531unieqi 4852 . . . . . . . . 9 𝐾 = (𝐽t 𝑌)
5453eqcomi 2747 . . . . . . . 8 (𝐽t 𝑌) = 𝐾
5554ntrss2 22208 . . . . . . 7 ((𝐾 ∈ Top ∧ 𝑆 (𝐽t 𝑌)) → ((int‘𝐾)‘𝑆) ⊆ 𝑆)
5651, 52, 55syl2anc 584 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) → ((int‘𝐾)‘𝑆) ⊆ 𝑆)
57 unss1 4113 . . . . . 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 3931 . . . 4 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) → 𝑜 ⊆ (𝑆 ∪ (𝑋𝑌)))
60 simpl1 1190 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽𝑜 ⊆ (𝑆 ∪ (𝑋𝑌)))) → 𝐽 ∈ Top)
61 sstr 3929 . . . . . . . . . . . . . 14 ((𝑆𝑌𝑌𝑋) → 𝑆𝑋)
6261ancoms 459 . . . . . . . . . . . . 13 ((𝑌𝑋𝑆𝑌) → 𝑆𝑋)
63623adant1 1129 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑆𝑋)
6463adantr 481 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽𝑜 ⊆ (𝑆 ∪ (𝑋𝑌)))) → 𝑆𝑋)
65 difss 4066 . . . . . . . . . . 11 (𝑋𝑌) ⊆ 𝑋
66 unss 4118 . . . . . . . . . . 11 ((𝑆𝑋 ∧ (𝑋𝑌) ⊆ 𝑋) ↔ (𝑆 ∪ (𝑋𝑌)) ⊆ 𝑋)
6764, 65, 66sylanblc 589 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽𝑜 ⊆ (𝑆 ∪ (𝑋𝑌)))) → (𝑆 ∪ (𝑋𝑌)) ⊆ 𝑋)
68 simprl 768 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽𝑜 ⊆ (𝑆 ∪ (𝑋𝑌)))) → 𝑜𝐽)
69 simprr 770 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽𝑜 ⊆ (𝑆 ∪ (𝑋𝑌)))) → 𝑜 ⊆ (𝑆 ∪ (𝑋𝑌)))
704ssntr 22209 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ (𝑆 ∪ (𝑋𝑌)) ⊆ 𝑋) ∧ (𝑜𝐽𝑜 ⊆ (𝑆 ∪ (𝑋𝑌)))) → 𝑜 ⊆ ((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))))
7160, 67, 68, 69, 70syl22anc 836 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽𝑜 ⊆ (𝑆 ∪ (𝑋𝑌)))) → 𝑜 ⊆ ((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))))
7271ssrind 4169 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽𝑜 ⊆ (𝑆 ∪ (𝑋𝑌)))) → (𝑜𝑌) ⊆ (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌))
73 sseq1 3946 . . . . . . . 8 (((int‘𝐾)‘𝑆) = (𝑜𝑌) → (((int‘𝐾)‘𝑆) ⊆ (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌) ↔ (𝑜𝑌) ⊆ (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌)))
7472, 73syl5ibrcom 246 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽𝑜 ⊆ (𝑆 ∪ (𝑋𝑌)))) → (((int‘𝐾)‘𝑆) = (𝑜𝑌) → ((int‘𝐾)‘𝑆) ⊆ (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌)))
7574expr 457 . . . . . 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 455 . . . 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 3220 . 2 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((int‘𝐾)‘𝑆) ⊆ (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌))
801, 11eqeltrid 2843 . . 3 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝐾 ∈ Top)
8183adant3 1131 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑌 ∈ V)
8263, 65, 66sylanblc 589 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (𝑆 ∪ (𝑋𝑌)) ⊆ 𝑋)
834ntropn 22200 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑆 ∪ (𝑋𝑌)) ⊆ 𝑋) → ((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∈ 𝐽)
8419, 82, 83syl2anc 584 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∈ 𝐽)
85 elrestr 17139 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌 ∈ V ∧ ((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∈ 𝐽) → (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌) ∈ (𝐽t 𝑌))
8619, 81, 84, 85syl3anc 1370 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌) ∈ (𝐽t 𝑌))
8786, 1eleqtrrdi 2850 . . 3 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌) ∈ 𝐾)
884ntrss2 22208 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑆 ∪ (𝑋𝑌)) ⊆ 𝑋) → ((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ⊆ (𝑆 ∪ (𝑋𝑌)))
8919, 82, 88syl2anc 584 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ⊆ (𝑆 ∪ (𝑋𝑌)))
9089ssrind 4169 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌) ⊆ ((𝑆 ∪ (𝑋𝑌)) ∩ 𝑌))
91 elin 3903 . . . . . . 7 (𝑥 ∈ ((𝑆 ∪ (𝑋𝑌)) ∩ 𝑌) ↔ (𝑥 ∈ (𝑆 ∪ (𝑋𝑌)) ∧ 𝑥𝑌))
92 elun 4083 . . . . . . . . 9 (𝑥 ∈ (𝑆 ∪ (𝑋𝑌)) ↔ (𝑥𝑆𝑥 ∈ (𝑋𝑌)))
93 orcom 867 . . . . . . . . . 10 ((𝑥𝑆𝑥 ∈ (𝑋𝑌)) ↔ (𝑥 ∈ (𝑋𝑌) ∨ 𝑥𝑆))
94 df-or 845 . . . . . . . . . 10 ((𝑥 ∈ (𝑋𝑌) ∨ 𝑥𝑆) ↔ (¬ 𝑥 ∈ (𝑋𝑌) → 𝑥𝑆))
9593, 94bitri 274 . . . . . . . . 9 ((𝑥𝑆𝑥 ∈ (𝑋𝑌)) ↔ (¬ 𝑥 ∈ (𝑋𝑌) → 𝑥𝑆))
9692, 95bitri 274 . . . . . . . 8 (𝑥 ∈ (𝑆 ∪ (𝑋𝑌)) ↔ (¬ 𝑥 ∈ (𝑋𝑌) → 𝑥𝑆))
9796anbi1i 624 . . . . . . 7 ((𝑥 ∈ (𝑆 ∪ (𝑋𝑌)) ∧ 𝑥𝑌) ↔ ((¬ 𝑥 ∈ (𝑋𝑌) → 𝑥𝑆) ∧ 𝑥𝑌))
9891, 97bitri 274 . . . . . 6 (𝑥 ∈ ((𝑆 ∪ (𝑋𝑌)) ∩ 𝑌) ↔ ((¬ 𝑥 ∈ (𝑋𝑌) → 𝑥𝑆) ∧ 𝑥𝑌))
99 elndif 4063 . . . . . . . . 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 408 . . . . . . 7 (((¬ 𝑥 ∈ (𝑋𝑌) → 𝑥𝑆) ∧ 𝑥𝑌) → 𝑥𝑆)
103102a1i 11 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (((¬ 𝑥 ∈ (𝑋𝑌) → 𝑥𝑆) ∧ 𝑥𝑌) → 𝑥𝑆))
10498, 103syl5bi 241 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (𝑥 ∈ ((𝑆 ∪ (𝑋𝑌)) ∩ 𝑌) → 𝑥𝑆))
105104ssrdv 3927 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((𝑆 ∪ (𝑋𝑌)) ∩ 𝑌) ⊆ 𝑆)
10690, 105sstrd 3931 . . 3 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌) ⊆ 𝑆)
10754ssntr 22209 . . 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 836 . 2 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌) ⊆ ((int‘𝐾)‘𝑆))
10979, 108eqssd 3938 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 205  wa 396  wo 844  w3a 1086   = wceq 1539  wcel 2106  wrex 3065  Vcvv 3432  cdif 3884  cun 3885  cin 3886  wss 3887   cuni 4839  cfv 6433  (class class class)co 7275  t crest 17131  Topctop 22042  intcnt 22168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-en 8734  df-fin 8737  df-fi 9170  df-rest 17133  df-topgen 17154  df-top 22043  df-topon 22060  df-bases 22096  df-ntr 22171
This theorem is referenced by:  llycmpkgen2  22701  dvreslem  25073  dvres2lem  25074  dvaddbr  25102  dvmulbr  25103  dvcnvrelem2  25182  limciccioolb  43162  limcicciooub  43178  ioccncflimc  43426  icocncflimc  43430  cncfiooicclem1  43434  fourierdlem62  43709
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