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Theorem restntr 21796
Description: An interior in a subspace topology. Willard in General Topology says that there is no analogue of restcls 21795 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 6666 . . . . . 6 (int‘𝐾) = (int‘(𝐽t 𝑌))
32fveq1i 6664 . . . . 5 ((int‘𝐾)‘𝑆) = ((int‘(𝐽t 𝑌))‘𝑆)
4 restcls.1 . . . . . . . . . 10 𝑋 = 𝐽
54topopn 21520 . . . . . . . . 9 (𝐽 ∈ Top → 𝑋𝐽)
6 ssexg 5214 . . . . . . . . . 10 ((𝑌𝑋𝑋𝐽) → 𝑌 ∈ V)
76ancoms 462 . . . . . . . . 9 ((𝑋𝐽𝑌𝑋) → 𝑌 ∈ V)
85, 7sylan 583 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑌𝑋) → 𝑌 ∈ V)
9 resttop 21774 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑌 ∈ V) → (𝐽t 𝑌) ∈ Top)
108, 9syldan 594 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑌𝑋) → (𝐽t 𝑌) ∈ Top)
11103adant3 1129 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (𝐽t 𝑌) ∈ Top)
124restuni 21776 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑌𝑋) → 𝑌 = (𝐽t 𝑌))
1312sseq2d 3985 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑌𝑋) → (𝑆𝑌𝑆 (𝐽t 𝑌)))
1413biimp3a 1466 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑆 (𝐽t 𝑌))
15 eqid 2824 . . . . . . 7 (𝐽t 𝑌) = (𝐽t 𝑌)
1615ntropn 21663 . . . . . 6 (((𝐽t 𝑌) ∈ Top ∧ 𝑆 (𝐽t 𝑌)) → ((int‘(𝐽t 𝑌))‘𝑆) ∈ (𝐽t 𝑌))
1711, 14, 16syl2anc 587 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((int‘(𝐽t 𝑌))‘𝑆) ∈ (𝐽t 𝑌))
183, 17eqeltrid 2920 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((int‘𝐾)‘𝑆) ∈ (𝐽t 𝑌))
19 simp1 1133 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝐽 ∈ Top)
20 uniexg 7462 . . . . . . . . 9 (𝐽 ∈ Top → 𝐽 ∈ V)
214, 20eqeltrid 2920 . . . . . . . 8 (𝐽 ∈ Top → 𝑋 ∈ V)
22 ssexg 5214 . . . . . . . 8 ((𝑌𝑋𝑋 ∈ V) → 𝑌 ∈ V)
2321, 22sylan2 595 . . . . . . 7 ((𝑌𝑋𝐽 ∈ Top) → 𝑌 ∈ V)
2423ancoms 462 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋) → 𝑌 ∈ V)
25243adant3 1129 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑌 ∈ V)
26 elrest 16703 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌 ∈ V) → (((int‘𝐾)‘𝑆) ∈ (𝐽t 𝑌) ↔ ∃𝑜𝐽 ((int‘𝐾)‘𝑆) = (𝑜𝑌)))
2719, 25, 26syl2anc 587 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (((int‘𝐾)‘𝑆) ∈ (𝐽t 𝑌) ↔ ∃𝑜𝐽 ((int‘𝐾)‘𝑆) = (𝑜𝑌)))
2818, 27mpbid 235 . . 3 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ∃𝑜𝐽 ((int‘𝐾)‘𝑆) = (𝑜𝑌))
294eltopss 21521 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑜𝐽) → 𝑜𝑋)
3029sseld 3952 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑜𝐽) → (𝑥𝑜𝑥𝑋))
3130adantrr 716 . . . . . . . . 9 ((𝐽 ∈ Top ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) → (𝑥𝑜𝑥𝑋))
32313ad2antl1 1182 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) → (𝑥𝑜𝑥𝑋))
33 eldif 3929 . . . . . . . . . 10 (𝑥 ∈ (𝑋𝑌) ↔ (𝑥𝑋 ∧ ¬ 𝑥𝑌))
3433simplbi2 504 . . . . . . . . 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 3935 . . . . . . . . . . 11 (𝑥 ∈ (𝑜𝑌) ↔ (𝑥𝑜𝑥𝑌))
38 eleq2 2904 . . . . . . . . . . . . 13 (((int‘𝐾)‘𝑆) = (𝑜𝑌) → (𝑥 ∈ ((int‘𝐾)‘𝑆) ↔ 𝑥 ∈ (𝑜𝑌)))
39 elun1 4138 . . . . . . . . . . . . 13 (𝑥 ∈ ((int‘𝐾)‘𝑆) → 𝑥 ∈ (((int‘𝐾)‘𝑆) ∪ (𝑋𝑌)))
4038, 39syl6bir 257 . . . . . . . . . . . 12 (((int‘𝐾)‘𝑆) = (𝑜𝑌) → (𝑥 ∈ (𝑜𝑌) → 𝑥 ∈ (((int‘𝐾)‘𝑆) ∪ (𝑋𝑌))))
4140ad2antll 728 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) → (𝑥 ∈ (𝑜𝑌) → 𝑥 ∈ (((int‘𝐾)‘𝑆) ∪ (𝑋𝑌))))
4237, 41syl5bir 246 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) → ((𝑥𝑜𝑥𝑌) → 𝑥 ∈ (((int‘𝐾)‘𝑆) ∪ (𝑋𝑌))))
4342expdimp 456 . . . . . . . . 9 ((((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) ∧ 𝑥𝑜) → (𝑥𝑌𝑥 ∈ (((int‘𝐾)‘𝑆) ∪ (𝑋𝑌))))
44 elun2 4139 . . . . . . . . . 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 416 . . . . . . 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 3959 . . . . 5 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) → 𝑜 ⊆ (((int‘𝐾)‘𝑆) ∪ (𝑋𝑌)))
5011adantr 484 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) → (𝐽t 𝑌) ∈ Top)
511, 50eqeltrid 2920 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) → 𝐾 ∈ Top)
5214adantr 484 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) → 𝑆 (𝐽t 𝑌))
531unieqi 4837 . . . . . . . . 9 𝐾 = (𝐽t 𝑌)
5453eqcomi 2833 . . . . . . . 8 (𝐽t 𝑌) = 𝐾
5554ntrss2 21671 . . . . . . 7 ((𝐾 ∈ Top ∧ 𝑆 (𝐽t 𝑌)) → ((int‘𝐾)‘𝑆) ⊆ 𝑆)
5651, 52, 55syl2anc 587 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) → ((int‘𝐾)‘𝑆) ⊆ 𝑆)
57 unss1 4141 . . . . . 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 3963 . . . 4 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜𝑌))) → 𝑜 ⊆ (𝑆 ∪ (𝑋𝑌)))
60 simpl1 1188 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽𝑜 ⊆ (𝑆 ∪ (𝑋𝑌)))) → 𝐽 ∈ Top)
61 sstr 3961 . . . . . . . . . . . . . 14 ((𝑆𝑌𝑌𝑋) → 𝑆𝑋)
6261ancoms 462 . . . . . . . . . . . . 13 ((𝑌𝑋𝑆𝑌) → 𝑆𝑋)
63623adant1 1127 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑆𝑋)
6463adantr 484 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽𝑜 ⊆ (𝑆 ∪ (𝑋𝑌)))) → 𝑆𝑋)
65 difss 4094 . . . . . . . . . . 11 (𝑋𝑌) ⊆ 𝑋
66 unss 4146 . . . . . . . . . . 11 ((𝑆𝑋 ∧ (𝑋𝑌) ⊆ 𝑋) ↔ (𝑆 ∪ (𝑋𝑌)) ⊆ 𝑋)
6764, 65, 66sylanblc 592 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽𝑜 ⊆ (𝑆 ∪ (𝑋𝑌)))) → (𝑆 ∪ (𝑋𝑌)) ⊆ 𝑋)
68 simprl 770 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽𝑜 ⊆ (𝑆 ∪ (𝑋𝑌)))) → 𝑜𝐽)
69 simprr 772 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽𝑜 ⊆ (𝑆 ∪ (𝑋𝑌)))) → 𝑜 ⊆ (𝑆 ∪ (𝑋𝑌)))
704ssntr 21672 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ (𝑆 ∪ (𝑋𝑌)) ⊆ 𝑋) ∧ (𝑜𝐽𝑜 ⊆ (𝑆 ∪ (𝑋𝑌)))) → 𝑜 ⊆ ((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))))
7160, 67, 68, 69, 70syl22anc 837 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽𝑜 ⊆ (𝑆 ∪ (𝑋𝑌)))) → 𝑜 ⊆ ((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))))
7271ssrind 4197 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽𝑜 ⊆ (𝑆 ∪ (𝑋𝑌)))) → (𝑜𝑌) ⊆ (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌))
73 sseq1 3978 . . . . . . . 8 (((int‘𝐾)‘𝑆) = (𝑜𝑌) → (((int‘𝐾)‘𝑆) ⊆ (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌) ↔ (𝑜𝑌) ⊆ (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌)))
7472, 73syl5ibrcom 250 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑜𝐽𝑜 ⊆ (𝑆 ∪ (𝑋𝑌)))) → (((int‘𝐾)‘𝑆) = (𝑜𝑌) → ((int‘𝐾)‘𝑆) ⊆ (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌)))
7574expr 460 . . . . . 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 458 . . . 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 3283 . 2 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((int‘𝐾)‘𝑆) ⊆ (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌))
801, 11eqeltrid 2920 . . 3 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝐾 ∈ Top)
8183adant3 1129 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑌 ∈ V)
8263, 65, 66sylanblc 592 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (𝑆 ∪ (𝑋𝑌)) ⊆ 𝑋)
834ntropn 21663 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑆 ∪ (𝑋𝑌)) ⊆ 𝑋) → ((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∈ 𝐽)
8419, 82, 83syl2anc 587 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∈ 𝐽)
85 elrestr 16704 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌 ∈ V ∧ ((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∈ 𝐽) → (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌) ∈ (𝐽t 𝑌))
8619, 81, 84, 85syl3anc 1368 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌) ∈ (𝐽t 𝑌))
8786, 1eleqtrrdi 2927 . . 3 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌) ∈ 𝐾)
884ntrss2 21671 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑆 ∪ (𝑋𝑌)) ⊆ 𝑋) → ((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ⊆ (𝑆 ∪ (𝑋𝑌)))
8919, 82, 88syl2anc 587 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ⊆ (𝑆 ∪ (𝑋𝑌)))
9089ssrind 4197 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌) ⊆ ((𝑆 ∪ (𝑋𝑌)) ∩ 𝑌))
91 elin 3935 . . . . . . 7 (𝑥 ∈ ((𝑆 ∪ (𝑋𝑌)) ∩ 𝑌) ↔ (𝑥 ∈ (𝑆 ∪ (𝑋𝑌)) ∧ 𝑥𝑌))
92 elun 4111 . . . . . . . . 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 278 . . . . . . . . 9 ((𝑥𝑆𝑥 ∈ (𝑋𝑌)) ↔ (¬ 𝑥 ∈ (𝑋𝑌) → 𝑥𝑆))
9692, 95bitri 278 . . . . . . . 8 (𝑥 ∈ (𝑆 ∪ (𝑋𝑌)) ↔ (¬ 𝑥 ∈ (𝑋𝑌) → 𝑥𝑆))
9796anbi1i 626 . . . . . . 7 ((𝑥 ∈ (𝑆 ∪ (𝑋𝑌)) ∧ 𝑥𝑌) ↔ ((¬ 𝑥 ∈ (𝑋𝑌) → 𝑥𝑆) ∧ 𝑥𝑌))
9891, 97bitri 278 . . . . . 6 (𝑥 ∈ ((𝑆 ∪ (𝑋𝑌)) ∩ 𝑌) ↔ ((¬ 𝑥 ∈ (𝑋𝑌) → 𝑥𝑆) ∧ 𝑥𝑌))
99 elndif 4091 . . . . . . . . 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 411 . . . . . . 7 (((¬ 𝑥 ∈ (𝑋𝑌) → 𝑥𝑆) ∧ 𝑥𝑌) → 𝑥𝑆)
103102a1i 11 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (((¬ 𝑥 ∈ (𝑋𝑌) → 𝑥𝑆) ∧ 𝑥𝑌) → 𝑥𝑆))
10498, 103syl5bi 245 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (𝑥 ∈ ((𝑆 ∪ (𝑋𝑌)) ∩ 𝑌) → 𝑥𝑆))
105104ssrdv 3959 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((𝑆 ∪ (𝑋𝑌)) ∩ 𝑌) ⊆ 𝑆)
10690, 105sstrd 3963 . . 3 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌) ⊆ 𝑆)
10754ssntr 21672 . . 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 837 . 2 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌) ⊆ ((int‘𝐾)‘𝑆))
10979, 108eqssd 3970 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 209  wa 399  wo 844  w3a 1084   = wceq 1538  wcel 2115  wrex 3134  Vcvv 3480  cdif 3916  cun 3917  cin 3918  wss 3919   cuni 4824  cfv 6345  (class class class)co 7151  t crest 16696  Topctop 21507  intcnt 21631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7457
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-pred 6137  df-ord 6183  df-on 6184  df-lim 6185  df-suc 6186  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fo 6351  df-f1o 6352  df-fv 6353  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7577  df-1st 7686  df-2nd 7687  df-wrecs 7945  df-recs 8006  df-rdg 8044  df-oadd 8104  df-er 8287  df-en 8508  df-fin 8511  df-fi 8874  df-rest 16698  df-topgen 16719  df-top 21508  df-topon 21525  df-bases 21560  df-ntr 21634
This theorem is referenced by:  llycmpkgen2  22164  dvreslem  24521  dvres2lem  24522  dvaddbr  24550  dvmulbr  24551  dvcnvrelem2  24630  limciccioolb  42216  limcicciooub  42232  ioccncflimc  42480  icocncflimc  42484  cncfiooicclem1  42488  fourierdlem62  42763
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