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Theorem ntrneik13 43670
Description: The interior of the intersection of any pair equals intersection of interiors if and only if the intersection of any pair belonging to the neighborhood of a point is equivalent to both of the pair belonging to the neighborhood of that point. (Contributed by RP, 19-Jun-2021.)
Hypotheses
Ref Expression
ntrnei.o 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
ntrnei.f 𝐹 = (𝒫 𝐵𝑂𝐵)
ntrnei.r (𝜑𝐼𝐹𝑁)
Assertion
Ref Expression
ntrneik13 (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠𝑡)) = ((𝐼𝑠) ∩ (𝐼𝑡)) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) ∈ (𝑁𝑥) ↔ (𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥)))))
Distinct variable groups:   𝐵,𝑖,𝑗,𝑘,𝑙,𝑚,𝑠,𝑡,𝑥   𝑘,𝐼,𝑙,𝑚,𝑥   𝜑,𝑖,𝑗,𝑘,𝑙,𝑠,𝑡,𝑥
Allowed substitution hints:   𝜑(𝑚)   𝐹(𝑥,𝑡,𝑖,𝑗,𝑘,𝑚,𝑠,𝑙)   𝐼(𝑡,𝑖,𝑗,𝑠)   𝑁(𝑥,𝑡,𝑖,𝑗,𝑘,𝑚,𝑠,𝑙)   𝑂(𝑥,𝑡,𝑖,𝑗,𝑘,𝑚,𝑠,𝑙)

Proof of Theorem ntrneik13
StepHypRef Expression
1 dfss3 3965 . . . . . . . . 9 ((𝐼‘(𝑠𝑡)) ⊆ ((𝐼𝑠) ∩ (𝐼𝑡)) ↔ ∀𝑥 ∈ (𝐼‘(𝑠𝑡))𝑥 ∈ ((𝐼𝑠) ∩ (𝐼𝑡)))
2 ntrnei.o . . . . . . . . . . . . . . 15 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
3 ntrnei.f . . . . . . . . . . . . . . 15 𝐹 = (𝒫 𝐵𝑂𝐵)
4 ntrnei.r . . . . . . . . . . . . . . 15 (𝜑𝐼𝐹𝑁)
52, 3, 4ntrneiiex 43648 . . . . . . . . . . . . . 14 (𝜑𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
6 elmapi 8868 . . . . . . . . . . . . . 14 (𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
75, 6syl 17 . . . . . . . . . . . . 13 (𝜑𝐼:𝒫 𝐵⟶𝒫 𝐵)
87ad2antrr 724 . . . . . . . . . . . 12 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
92, 3, 4ntrneibex 43645 . . . . . . . . . . . . . 14 (𝜑𝐵 ∈ V)
109ad2antrr 724 . . . . . . . . . . . . 13 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝐵 ∈ V)
11 simplr 767 . . . . . . . . . . . . . 14 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
12 elpwi 4611 . . . . . . . . . . . . . 14 (𝑠 ∈ 𝒫 𝐵𝑠𝐵)
13 ssinss1 4236 . . . . . . . . . . . . . 14 (𝑠𝐵 → (𝑠𝑡) ⊆ 𝐵)
1411, 12, 133syl 18 . . . . . . . . . . . . 13 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝑠𝑡) ⊆ 𝐵)
1510, 14sselpwd 5329 . . . . . . . . . . . 12 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝑠𝑡) ∈ 𝒫 𝐵)
168, 15ffvelcdmd 7094 . . . . . . . . . . 11 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐼‘(𝑠𝑡)) ∈ 𝒫 𝐵)
1716elpwid 4613 . . . . . . . . . 10 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐼‘(𝑠𝑡)) ⊆ 𝐵)
18 ralss 4051 . . . . . . . . . 10 ((𝐼‘(𝑠𝑡)) ⊆ 𝐵 → (∀𝑥 ∈ (𝐼‘(𝑠𝑡))𝑥 ∈ ((𝐼𝑠) ∩ (𝐼𝑡)) ↔ ∀𝑥𝐵 (𝑥 ∈ (𝐼‘(𝑠𝑡)) → 𝑥 ∈ ((𝐼𝑠) ∩ (𝐼𝑡)))))
1917, 18syl 17 . . . . . . . . 9 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (∀𝑥 ∈ (𝐼‘(𝑠𝑡))𝑥 ∈ ((𝐼𝑠) ∩ (𝐼𝑡)) ↔ ∀𝑥𝐵 (𝑥 ∈ (𝐼‘(𝑠𝑡)) → 𝑥 ∈ ((𝐼𝑠) ∩ (𝐼𝑡)))))
201, 19bitrid 282 . . . . . . . 8 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → ((𝐼‘(𝑠𝑡)) ⊆ ((𝐼𝑠) ∩ (𝐼𝑡)) ↔ ∀𝑥𝐵 (𝑥 ∈ (𝐼‘(𝑠𝑡)) → 𝑥 ∈ ((𝐼𝑠) ∩ (𝐼𝑡)))))
21 dfss3 3965 . . . . . . . . 9 (((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ (𝐼‘(𝑠𝑡)) ↔ ∀𝑥 ∈ ((𝐼𝑠) ∩ (𝐼𝑡))𝑥 ∈ (𝐼‘(𝑠𝑡)))
227ffvelcdmda 7093 . . . . . . . . . . . . 13 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐼𝑠) ∈ 𝒫 𝐵)
2322elpwid 4613 . . . . . . . . . . . 12 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐼𝑠) ⊆ 𝐵)
24 ssinss1 4236 . . . . . . . . . . . 12 ((𝐼𝑠) ⊆ 𝐵 → ((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ 𝐵)
2523, 24syl 17 . . . . . . . . . . 11 ((𝜑𝑠 ∈ 𝒫 𝐵) → ((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ 𝐵)
2625adantr 479 . . . . . . . . . 10 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → ((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ 𝐵)
27 ralss 4051 . . . . . . . . . 10 (((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ 𝐵 → (∀𝑥 ∈ ((𝐼𝑠) ∩ (𝐼𝑡))𝑥 ∈ (𝐼‘(𝑠𝑡)) ↔ ∀𝑥𝐵 (𝑥 ∈ ((𝐼𝑠) ∩ (𝐼𝑡)) → 𝑥 ∈ (𝐼‘(𝑠𝑡)))))
2826, 27syl 17 . . . . . . . . 9 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (∀𝑥 ∈ ((𝐼𝑠) ∩ (𝐼𝑡))𝑥 ∈ (𝐼‘(𝑠𝑡)) ↔ ∀𝑥𝐵 (𝑥 ∈ ((𝐼𝑠) ∩ (𝐼𝑡)) → 𝑥 ∈ (𝐼‘(𝑠𝑡)))))
2921, 28bitrid 282 . . . . . . . 8 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ (𝐼‘(𝑠𝑡)) ↔ ∀𝑥𝐵 (𝑥 ∈ ((𝐼𝑠) ∩ (𝐼𝑡)) → 𝑥 ∈ (𝐼‘(𝑠𝑡)))))
3020, 29anbi12d 630 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (((𝐼‘(𝑠𝑡)) ⊆ ((𝐼𝑠) ∩ (𝐼𝑡)) ∧ ((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ (𝐼‘(𝑠𝑡))) ↔ (∀𝑥𝐵 (𝑥 ∈ (𝐼‘(𝑠𝑡)) → 𝑥 ∈ ((𝐼𝑠) ∩ (𝐼𝑡))) ∧ ∀𝑥𝐵 (𝑥 ∈ ((𝐼𝑠) ∩ (𝐼𝑡)) → 𝑥 ∈ (𝐼‘(𝑠𝑡))))))
31 eqss 3992 . . . . . . 7 ((𝐼‘(𝑠𝑡)) = ((𝐼𝑠) ∩ (𝐼𝑡)) ↔ ((𝐼‘(𝑠𝑡)) ⊆ ((𝐼𝑠) ∩ (𝐼𝑡)) ∧ ((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ (𝐼‘(𝑠𝑡))))
32 ralbiim 3102 . . . . . . 7 (∀𝑥𝐵 (𝑥 ∈ (𝐼‘(𝑠𝑡)) ↔ 𝑥 ∈ ((𝐼𝑠) ∩ (𝐼𝑡))) ↔ (∀𝑥𝐵 (𝑥 ∈ (𝐼‘(𝑠𝑡)) → 𝑥 ∈ ((𝐼𝑠) ∩ (𝐼𝑡))) ∧ ∀𝑥𝐵 (𝑥 ∈ ((𝐼𝑠) ∩ (𝐼𝑡)) → 𝑥 ∈ (𝐼‘(𝑠𝑡)))))
3330, 31, 323bitr4g 313 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → ((𝐼‘(𝑠𝑡)) = ((𝐼𝑠) ∩ (𝐼𝑡)) ↔ ∀𝑥𝐵 (𝑥 ∈ (𝐼‘(𝑠𝑡)) ↔ 𝑥 ∈ ((𝐼𝑠) ∩ (𝐼𝑡)))))
344ad3antrrr 728 . . . . . . . . 9 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝐼𝐹𝑁)
35 simpr 483 . . . . . . . . 9 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝑥𝐵)
369adantr 479 . . . . . . . . . . 11 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝐵 ∈ V)
37 simpr 483 . . . . . . . . . . . . 13 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
3837elpwid 4613 . . . . . . . . . . . 12 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝑠𝐵)
3938, 13syl 17 . . . . . . . . . . 11 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝑠𝑡) ⊆ 𝐵)
4036, 39sselpwd 5329 . . . . . . . . . 10 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝑠𝑡) ∈ 𝒫 𝐵)
4140ad2antrr 724 . . . . . . . . 9 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → (𝑠𝑡) ∈ 𝒫 𝐵)
422, 3, 34, 35, 41ntrneiel 43653 . . . . . . . 8 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → (𝑥 ∈ (𝐼‘(𝑠𝑡)) ↔ (𝑠𝑡) ∈ (𝑁𝑥)))
43 elin 3960 . . . . . . . . 9 (𝑥 ∈ ((𝐼𝑠) ∩ (𝐼𝑡)) ↔ (𝑥 ∈ (𝐼𝑠) ∧ 𝑥 ∈ (𝐼𝑡)))
44 simpllr 774 . . . . . . . . . . 11 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝑠 ∈ 𝒫 𝐵)
452, 3, 34, 35, 44ntrneiel 43653 . . . . . . . . . 10 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → (𝑥 ∈ (𝐼𝑠) ↔ 𝑠 ∈ (𝑁𝑥)))
46 simplr 767 . . . . . . . . . . 11 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝑡 ∈ 𝒫 𝐵)
472, 3, 34, 35, 46ntrneiel 43653 . . . . . . . . . 10 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → (𝑥 ∈ (𝐼𝑡) ↔ 𝑡 ∈ (𝑁𝑥)))
4845, 47anbi12d 630 . . . . . . . . 9 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → ((𝑥 ∈ (𝐼𝑠) ∧ 𝑥 ∈ (𝐼𝑡)) ↔ (𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥))))
4943, 48bitrid 282 . . . . . . . 8 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → (𝑥 ∈ ((𝐼𝑠) ∩ (𝐼𝑡)) ↔ (𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥))))
5042, 49bibi12d 344 . . . . . . 7 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → ((𝑥 ∈ (𝐼‘(𝑠𝑡)) ↔ 𝑥 ∈ ((𝐼𝑠) ∩ (𝐼𝑡))) ↔ ((𝑠𝑡) ∈ (𝑁𝑥) ↔ (𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥)))))
5150ralbidva 3165 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (∀𝑥𝐵 (𝑥 ∈ (𝐼‘(𝑠𝑡)) ↔ 𝑥 ∈ ((𝐼𝑠) ∩ (𝐼𝑡))) ↔ ∀𝑥𝐵 ((𝑠𝑡) ∈ (𝑁𝑥) ↔ (𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥)))))
5233, 51bitrd 278 . . . . 5 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → ((𝐼‘(𝑠𝑡)) = ((𝐼𝑠) ∩ (𝐼𝑡)) ↔ ∀𝑥𝐵 ((𝑠𝑡) ∈ (𝑁𝑥) ↔ (𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥)))))
5352ralbidva 3165 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵) → (∀𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠𝑡)) = ((𝐼𝑠) ∩ (𝐼𝑡)) ↔ ∀𝑡 ∈ 𝒫 𝐵𝑥𝐵 ((𝑠𝑡) ∈ (𝑁𝑥) ↔ (𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥)))))
54 ralcom 3276 . . . 4 (∀𝑡 ∈ 𝒫 𝐵𝑥𝐵 ((𝑠𝑡) ∈ (𝑁𝑥) ↔ (𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥))) ↔ ∀𝑥𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) ∈ (𝑁𝑥) ↔ (𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥))))
5553, 54bitrdi 286 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵) → (∀𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠𝑡)) = ((𝐼𝑠) ∩ (𝐼𝑡)) ↔ ∀𝑥𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) ∈ (𝑁𝑥) ↔ (𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥)))))
5655ralbidva 3165 . 2 (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠𝑡)) = ((𝐼𝑠) ∩ (𝐼𝑡)) ↔ ∀𝑠 ∈ 𝒫 𝐵𝑥𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) ∈ (𝑁𝑥) ↔ (𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥)))))
57 ralcom 3276 . 2 (∀𝑠 ∈ 𝒫 𝐵𝑥𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) ∈ (𝑁𝑥) ↔ (𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥))) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) ∈ (𝑁𝑥) ↔ (𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥))))
5856, 57bitrdi 286 1 (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠𝑡)) = ((𝐼𝑠) ∩ (𝐼𝑡)) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) ∈ (𝑁𝑥) ↔ (𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wral 3050  {crab 3418  Vcvv 3461  cin 3943  wss 3944  𝒫 cpw 4604   class class class wbr 5149  cmpt 5232  wf 6545  cfv 6549  (class class class)co 7419  cmpo 7421  m cmap 8845
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  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-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7994  df-2nd 7995  df-map 8847
This theorem is referenced by: (None)
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