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Theorem restntr 20905
Description: An interior in a subspace topology. Willard in General Topology says that there is no analogue of restcls 20904 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 6156 . . . . . 6 (int‘𝐾) = (int‘(𝐽t 𝑌))
32fveq1i 6154 . . . . 5 ((int‘𝐾)‘𝑆) = ((int‘(𝐽t 𝑌))‘𝑆)
4 restcls.1 . . . . . . . . . 10 𝑋 = 𝐽
54topopn 20639 . . . . . . . . 9 (𝐽 ∈ Top → 𝑋𝐽)
6 ssexg 4769 . . . . . . . . . 10 ((𝑌𝑋𝑋𝐽) → 𝑌 ∈ V)
76ancoms 469 . . . . . . . . 9 ((𝑋𝐽𝑌𝑋) → 𝑌 ∈ V)
85, 7sylan 488 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑌𝑋) → 𝑌 ∈ V)
9 resttop 20883 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑌 ∈ V) → (𝐽t 𝑌) ∈ Top)
108, 9syldan 487 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑌𝑋) → (𝐽t 𝑌) ∈ Top)
11103adant3 1079 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (𝐽t 𝑌) ∈ Top)
124restuni 20885 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑌𝑋) → 𝑌 = (𝐽t 𝑌))
1312sseq2d 3617 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑌𝑋) → (𝑆𝑌𝑆 (𝐽t 𝑌)))
1413biimp3a 1429 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑆 (𝐽t 𝑌))
15 eqid 2621 . . . . . . 7 (𝐽t 𝑌) = (𝐽t 𝑌)
1615ntropn 20772 . . . . . 6 (((𝐽t 𝑌) ∈ Top ∧ 𝑆 (𝐽t 𝑌)) → ((int‘(𝐽t 𝑌))‘𝑆) ∈ (𝐽t 𝑌))
1711, 14, 16syl2anc 692 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((int‘(𝐽t 𝑌))‘𝑆) ∈ (𝐽t 𝑌))
183, 17syl5eqel 2702 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((int‘𝐾)‘𝑆) ∈ (𝐽t 𝑌))
19 simp1 1059 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝐽 ∈ Top)
20 uniexg 6915 . . . . . . . . 9 (𝐽 ∈ Top → 𝐽 ∈ V)
214, 20syl5eqel 2702 . . . . . . . 8 (𝐽 ∈ Top → 𝑋 ∈ V)
22 ssexg 4769 . . . . . . . 8 ((𝑌𝑋𝑋 ∈ V) → 𝑌 ∈ V)
2321, 22sylan2 491 . . . . . . 7 ((𝑌𝑋𝐽 ∈ Top) → 𝑌 ∈ V)
2423ancoms 469 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋) → 𝑌 ∈ V)
25243adant3 1079 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑌 ∈ V)
26 elrest 16016 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌 ∈ V) → (((int‘𝐾)‘𝑆) ∈ (𝐽t 𝑌) ↔ ∃𝑜𝐽 ((int‘𝐾)‘𝑆) = (𝑜𝑌)))
2719, 25, 26syl2anc 692 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (((int‘𝐾)‘𝑆) ∈ (𝐽t 𝑌) ↔ ∃𝑜𝐽 ((int‘𝐾)‘𝑆) = (𝑜𝑌)))
2818, 27mpbid 222 . . 3 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ∃𝑜𝐽 ((int‘𝐾)‘𝑆) = (𝑜𝑌))
294eltopss 20640 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑜𝐽) → 𝑜𝑋)
3029sseld 3586 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑜𝐽) → (𝑥𝑜𝑥𝑋))
3130adantrr 752 . . . . . . . . 9 ((𝐽 ∈ Top ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) → (𝑥𝑜𝑥𝑋))
32313ad2antl1 1221 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) → (𝑥𝑜𝑥𝑋))
33 eldif 3569 . . . . . . . . . 10 (𝑥 ∈ (𝑋𝑌) ↔ (𝑥𝑋 ∧ ¬ 𝑥𝑌))
3433simplbi2 654 . . . . . . . . 9 (𝑥𝑋 → (¬ 𝑥𝑌𝑥 ∈ (𝑋𝑌)))
3534orrd 393 . . . . . . . 8 (𝑥𝑋 → (𝑥𝑌𝑥 ∈ (𝑋𝑌)))
3632, 35syl6 35 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) → (𝑥𝑜 → (𝑥𝑌𝑥 ∈ (𝑋𝑌))))
37 elin 3779 . . . . . . . . . . 11 (𝑥 ∈ (𝑜𝑌) ↔ (𝑥𝑜𝑥𝑌))
38 eleq2 2687 . . . . . . . . . . . . 13 (((int‘𝐾)‘𝑆) = (𝑜𝑌) → (𝑥 ∈ ((int‘𝐾)‘𝑆) ↔ 𝑥 ∈ (𝑜𝑌)))
39 elun1 3763 . . . . . . . . . . . . 13 (𝑥 ∈ ((int‘𝐾)‘𝑆) → 𝑥 ∈ (((int‘𝐾)‘𝑆) ∪ (𝑋𝑌)))
4038, 39syl6bir 244 . . . . . . . . . . . 12 (((int‘𝐾)‘𝑆) = (𝑜𝑌) → (𝑥 ∈ (𝑜𝑌) → 𝑥 ∈ (((int‘𝐾)‘𝑆) ∪ (𝑋𝑌))))
4140ad2antll 764 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) → (𝑥 ∈ (𝑜𝑌) → 𝑥 ∈ (((int‘𝐾)‘𝑆) ∪ (𝑋𝑌))))
4237, 41syl5bir 233 . . . . . . . . . 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 3764 . . . . . . . . . 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 395 . . . . . . . 8 ((((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) ∧ 𝑥𝑜) → ((𝑥𝑌𝑥 ∈ (𝑋𝑌)) → 𝑥 ∈ (((int‘𝐾)‘𝑆) ∪ (𝑋𝑌))))
4746ex 450 . . . . . . 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 3593 . . . . 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, 50syl5eqel 2702 . . . . . . 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 4416 . . . . . . . . 9 𝐾 = (𝐽t 𝑌)
5453eqcomi 2630 . . . . . . . 8 (𝐽t 𝑌) = 𝐾
5554ntrss2 20780 . . . . . . 7 ((𝐾 ∈ Top ∧ 𝑆 (𝐽t 𝑌)) → ((int‘𝐾)‘𝑆) ⊆ 𝑆)
5651, 52, 55syl2anc 692 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) → ((int‘𝐾)‘𝑆) ⊆ 𝑆)
57 unss1 3765 . . . . . 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 3597 . . . 4 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) → 𝑜 ⊆ (𝑆 ∪ (𝑋𝑌)))
60 simpl1 1062 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽𝑜 ⊆ (𝑆 ∪ (𝑋𝑌)))) → 𝐽 ∈ Top)
61 sstr 3595 . . . . . . . . . . . . . 14 ((𝑆𝑌𝑌𝑋) → 𝑆𝑋)
6261ancoms 469 . . . . . . . . . . . . 13 ((𝑌𝑋𝑆𝑌) → 𝑆𝑋)
63623adant1 1077 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑆𝑋)
6463adantr 481 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽𝑜 ⊆ (𝑆 ∪ (𝑋𝑌)))) → 𝑆𝑋)
65 difss 3720 . . . . . . . . . . 11 (𝑋𝑌) ⊆ 𝑋
66 unss 3770 . . . . . . . . . . 11 ((𝑆𝑋 ∧ (𝑋𝑌) ⊆ 𝑋) ↔ (𝑆 ∪ (𝑋𝑌)) ⊆ 𝑋)
6764, 65, 66sylanblc 695 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽𝑜 ⊆ (𝑆 ∪ (𝑋𝑌)))) → (𝑆 ∪ (𝑋𝑌)) ⊆ 𝑋)
68 simprl 793 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽𝑜 ⊆ (𝑆 ∪ (𝑋𝑌)))) → 𝑜𝐽)
69 simprr 795 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽𝑜 ⊆ (𝑆 ∪ (𝑋𝑌)))) → 𝑜 ⊆ (𝑆 ∪ (𝑋𝑌)))
704ssntr 20781 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ (𝑆 ∪ (𝑋𝑌)) ⊆ 𝑋) ∧ (𝑜𝐽𝑜 ⊆ (𝑆 ∪ (𝑋𝑌)))) → 𝑜 ⊆ ((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))))
7160, 67, 68, 69, 70syl22anc 1324 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽𝑜 ⊆ (𝑆 ∪ (𝑋𝑌)))) → 𝑜 ⊆ ((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))))
72 ssrin 3821 . . . . . . . . 9 (𝑜 ⊆ ((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) → (𝑜𝑌) ⊆ (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌))
7371, 72syl 17 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽𝑜 ⊆ (𝑆 ∪ (𝑋𝑌)))) → (𝑜𝑌) ⊆ (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌))
74 sseq1 3610 . . . . . . . 8 (((int‘𝐾)‘𝑆) = (𝑜𝑌) → (((int‘𝐾)‘𝑆) ⊆ (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌) ↔ (𝑜𝑌) ⊆ (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌)))
7573, 74syl5ibrcom 237 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽𝑜 ⊆ (𝑆 ∪ (𝑋𝑌)))) → (((int‘𝐾)‘𝑆) = (𝑜𝑌) → ((int‘𝐾)‘𝑆) ⊆ (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌)))
7675expr 642 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ 𝑜𝐽) → (𝑜 ⊆ (𝑆 ∪ (𝑋𝑌)) → (((int‘𝐾)‘𝑆) = (𝑜𝑌) → ((int‘𝐾)‘𝑆) ⊆ (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌))))
7776com23 86 . . . . 5 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ 𝑜𝐽) → (((int‘𝐾)‘𝑆) = (𝑜𝑌) → (𝑜 ⊆ (𝑆 ∪ (𝑋𝑌)) → ((int‘𝐾)‘𝑆) ⊆ (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌))))
7877impr 648 . . . 4 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) → (𝑜 ⊆ (𝑆 ∪ (𝑋𝑌)) → ((int‘𝐾)‘𝑆) ⊆ (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌)))
7959, 78mpd 15 . . 3 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) → ((int‘𝐾)‘𝑆) ⊆ (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌))
8028, 79rexlimddv 3029 . 2 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((int‘𝐾)‘𝑆) ⊆ (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌))
811, 11syl5eqel 2702 . . 3 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝐾 ∈ Top)
8283adant3 1079 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑌 ∈ V)
8363, 65, 66sylanblc 695 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (𝑆 ∪ (𝑋𝑌)) ⊆ 𝑋)
844ntropn 20772 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑆 ∪ (𝑋𝑌)) ⊆ 𝑋) → ((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∈ 𝐽)
8519, 83, 84syl2anc 692 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∈ 𝐽)
86 elrestr 16017 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌 ∈ V ∧ ((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∈ 𝐽) → (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌) ∈ (𝐽t 𝑌))
8719, 82, 85, 86syl3anc 1323 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌) ∈ (𝐽t 𝑌))
8887, 1syl6eleqr 2709 . . 3 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌) ∈ 𝐾)
894ntrss2 20780 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑆 ∪ (𝑋𝑌)) ⊆ 𝑋) → ((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ⊆ (𝑆 ∪ (𝑋𝑌)))
9019, 83, 89syl2anc 692 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ⊆ (𝑆 ∪ (𝑋𝑌)))
91 ssrin 3821 . . . . 5 (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ⊆ (𝑆 ∪ (𝑋𝑌)) → (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌) ⊆ ((𝑆 ∪ (𝑋𝑌)) ∩ 𝑌))
9290, 91syl 17 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌) ⊆ ((𝑆 ∪ (𝑋𝑌)) ∩ 𝑌))
93 elin 3779 . . . . . . 7 (𝑥 ∈ ((𝑆 ∪ (𝑋𝑌)) ∩ 𝑌) ↔ (𝑥 ∈ (𝑆 ∪ (𝑋𝑌)) ∧ 𝑥𝑌))
94 elun 3736 . . . . . . . . 9 (𝑥 ∈ (𝑆 ∪ (𝑋𝑌)) ↔ (𝑥𝑆𝑥 ∈ (𝑋𝑌)))
95 orcom 402 . . . . . . . . . 10 ((𝑥𝑆𝑥 ∈ (𝑋𝑌)) ↔ (𝑥 ∈ (𝑋𝑌) ∨ 𝑥𝑆))
96 df-or 385 . . . . . . . . . 10 ((𝑥 ∈ (𝑋𝑌) ∨ 𝑥𝑆) ↔ (¬ 𝑥 ∈ (𝑋𝑌) → 𝑥𝑆))
9795, 96bitri 264 . . . . . . . . 9 ((𝑥𝑆𝑥 ∈ (𝑋𝑌)) ↔ (¬ 𝑥 ∈ (𝑋𝑌) → 𝑥𝑆))
9894, 97bitri 264 . . . . . . . 8 (𝑥 ∈ (𝑆 ∪ (𝑋𝑌)) ↔ (¬ 𝑥 ∈ (𝑋𝑌) → 𝑥𝑆))
9998anbi1i 730 . . . . . . 7 ((𝑥 ∈ (𝑆 ∪ (𝑋𝑌)) ∧ 𝑥𝑌) ↔ ((¬ 𝑥 ∈ (𝑋𝑌) → 𝑥𝑆) ∧ 𝑥𝑌))
10093, 99bitri 264 . . . . . 6 (𝑥 ∈ ((𝑆 ∪ (𝑋𝑌)) ∩ 𝑌) ↔ ((¬ 𝑥 ∈ (𝑋𝑌) → 𝑥𝑆) ∧ 𝑥𝑌))
101 elndif 3717 . . . . . . . . 9 (𝑥𝑌 → ¬ 𝑥 ∈ (𝑋𝑌))
102 pm2.27 42 . . . . . . . . 9 𝑥 ∈ (𝑋𝑌) → ((¬ 𝑥 ∈ (𝑋𝑌) → 𝑥𝑆) → 𝑥𝑆))
103101, 102syl 17 . . . . . . . 8 (𝑥𝑌 → ((¬ 𝑥 ∈ (𝑋𝑌) → 𝑥𝑆) → 𝑥𝑆))
104103impcom 446 . . . . . . 7 (((¬ 𝑥 ∈ (𝑋𝑌) → 𝑥𝑆) ∧ 𝑥𝑌) → 𝑥𝑆)
105104a1i 11 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (((¬ 𝑥 ∈ (𝑋𝑌) → 𝑥𝑆) ∧ 𝑥𝑌) → 𝑥𝑆))
106100, 105syl5bi 232 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (𝑥 ∈ ((𝑆 ∪ (𝑋𝑌)) ∩ 𝑌) → 𝑥𝑆))
107106ssrdv 3593 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((𝑆 ∪ (𝑋𝑌)) ∩ 𝑌) ⊆ 𝑆)
10892, 107sstrd 3597 . . 3 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌) ⊆ 𝑆)
10954ssntr 20781 . . 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))
11081, 14, 88, 108, 109syl22anc 1324 . 2 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌) ⊆ ((int‘𝐾)‘𝑆))
11180, 110eqssd 3604 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 196  wo 383  wa 384  w3a 1036   = wceq 1480  wcel 1987  wrex 2908  Vcvv 3189  cdif 3556  cun 3557  cin 3558  wss 3559   cuni 4407  cfv 5852  (class class class)co 6610  t crest 16009  Topctop 20626  intcnt 20740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-oadd 7516  df-er 7694  df-en 7907  df-fin 7910  df-fi 8268  df-rest 16011  df-topgen 16032  df-top 20627  df-topon 20644  df-bases 20670  df-ntr 20743
This theorem is referenced by:  llycmpkgen2  21272  dvreslem  23592  dvres2lem  23593  dvaddbr  23620  dvmulbr  23621  dvcnvrelem2  23698  limciccioolb  39280  limcicciooub  39296  ioccncflimc  39424  icocncflimc  39428  cncfiooicclem1  39432  fourierdlem62  39713
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