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Theorem ntrneik13 39231
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 3816 . . . . . . . . 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 39209 . . . . . . . . . . . . . 14 (𝜑𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
6 elmapi 8149 . . . . . . . . . . . . . 14 (𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
75, 6syl 17 . . . . . . . . . . . . 13 (𝜑𝐼:𝒫 𝐵⟶𝒫 𝐵)
87ad2antrr 717 . . . . . . . . . . . 12 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
92, 3, 4ntrneibex 39206 . . . . . . . . . . . . . 14 (𝜑𝐵 ∈ V)
109ad2antrr 717 . . . . . . . . . . . . 13 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝐵 ∈ V)
11 simplr 785 . . . . . . . . . . . . . 14 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
12 elpwi 4390 . . . . . . . . . . . . . 14 (𝑠 ∈ 𝒫 𝐵𝑠𝐵)
13 ssinss1 4068 . . . . . . . . . . . . . 14 (𝑠𝐵 → (𝑠𝑡) ⊆ 𝐵)
1411, 12, 133syl 18 . . . . . . . . . . . . 13 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝑠𝑡) ⊆ 𝐵)
1510, 14sselpwd 5034 . . . . . . . . . . . 12 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝑠𝑡) ∈ 𝒫 𝐵)
168, 15ffvelrnd 6614 . . . . . . . . . . 11 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐼‘(𝑠𝑡)) ∈ 𝒫 𝐵)
1716elpwid 4392 . . . . . . . . . 10 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐼‘(𝑠𝑡)) ⊆ 𝐵)
18 ralss 3895 . . . . . . . . . 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, 19syl5bb 275 . . . . . . . 8 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → ((𝐼‘(𝑠𝑡)) ⊆ ((𝐼𝑠) ∩ (𝐼𝑡)) ↔ ∀𝑥𝐵 (𝑥 ∈ (𝐼‘(𝑠𝑡)) → 𝑥 ∈ ((𝐼𝑠) ∩ (𝐼𝑡)))))
21 dfss3 3816 . . . . . . . . 9 (((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ (𝐼‘(𝑠𝑡)) ↔ ∀𝑥 ∈ ((𝐼𝑠) ∩ (𝐼𝑡))𝑥 ∈ (𝐼‘(𝑠𝑡)))
227ffvelrnda 6613 . . . . . . . . . . . . 13 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐼𝑠) ∈ 𝒫 𝐵)
2322elpwid 4392 . . . . . . . . . . . 12 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐼𝑠) ⊆ 𝐵)
24 ssinss1 4068 . . . . . . . . . . . 12 ((𝐼𝑠) ⊆ 𝐵 → ((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ 𝐵)
2523, 24syl 17 . . . . . . . . . . 11 ((𝜑𝑠 ∈ 𝒫 𝐵) → ((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ 𝐵)
2625adantr 474 . . . . . . . . . 10 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → ((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ 𝐵)
27 ralss 3895 . . . . . . . . . 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, 28syl5bb 275 . . . . . . . 8 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ (𝐼‘(𝑠𝑡)) ↔ ∀𝑥𝐵 (𝑥 ∈ ((𝐼𝑠) ∩ (𝐼𝑡)) → 𝑥 ∈ (𝐼‘(𝑠𝑡)))))
3020, 29anbi12d 624 . . . . . . 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 3842 . . . . . . 7 ((𝐼‘(𝑠𝑡)) = ((𝐼𝑠) ∩ (𝐼𝑡)) ↔ ((𝐼‘(𝑠𝑡)) ⊆ ((𝐼𝑠) ∩ (𝐼𝑡)) ∧ ((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ (𝐼‘(𝑠𝑡))))
32 ralbiim 3279 . . . . . . 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 306 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → ((𝐼‘(𝑠𝑡)) = ((𝐼𝑠) ∩ (𝐼𝑡)) ↔ ∀𝑥𝐵 (𝑥 ∈ (𝐼‘(𝑠𝑡)) ↔ 𝑥 ∈ ((𝐼𝑠) ∩ (𝐼𝑡)))))
344ad3antrrr 721 . . . . . . . . 9 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝐼𝐹𝑁)
35 simpr 479 . . . . . . . . 9 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝑥𝐵)
369adantr 474 . . . . . . . . . . 11 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝐵 ∈ V)
37 simpr 479 . . . . . . . . . . . . 13 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
3837elpwid 4392 . . . . . . . . . . . 12 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝑠𝐵)
3938, 13syl 17 . . . . . . . . . . 11 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝑠𝑡) ⊆ 𝐵)
4036, 39sselpwd 5034 . . . . . . . . . 10 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝑠𝑡) ∈ 𝒫 𝐵)
4140ad2antrr 717 . . . . . . . . 9 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → (𝑠𝑡) ∈ 𝒫 𝐵)
422, 3, 34, 35, 41ntrneiel 39214 . . . . . . . 8 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → (𝑥 ∈ (𝐼‘(𝑠𝑡)) ↔ (𝑠𝑡) ∈ (𝑁𝑥)))
43 elin 4025 . . . . . . . . 9 (𝑥 ∈ ((𝐼𝑠) ∩ (𝐼𝑡)) ↔ (𝑥 ∈ (𝐼𝑠) ∧ 𝑥 ∈ (𝐼𝑡)))
44 simpllr 793 . . . . . . . . . . 11 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝑠 ∈ 𝒫 𝐵)
452, 3, 34, 35, 44ntrneiel 39214 . . . . . . . . . 10 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → (𝑥 ∈ (𝐼𝑠) ↔ 𝑠 ∈ (𝑁𝑥)))
46 simplr 785 . . . . . . . . . . 11 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝑡 ∈ 𝒫 𝐵)
472, 3, 34, 35, 46ntrneiel 39214 . . . . . . . . . 10 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → (𝑥 ∈ (𝐼𝑡) ↔ 𝑡 ∈ (𝑁𝑥)))
4845, 47anbi12d 624 . . . . . . . . 9 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → ((𝑥 ∈ (𝐼𝑠) ∧ 𝑥 ∈ (𝐼𝑡)) ↔ (𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥))))
4943, 48syl5bb 275 . . . . . . . 8 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → (𝑥 ∈ ((𝐼𝑠) ∩ (𝐼𝑡)) ↔ (𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥))))
5042, 49bibi12d 337 . . . . . . 7 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → ((𝑥 ∈ (𝐼‘(𝑠𝑡)) ↔ 𝑥 ∈ ((𝐼𝑠) ∩ (𝐼𝑡))) ↔ ((𝑠𝑡) ∈ (𝑁𝑥) ↔ (𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥)))))
5150ralbidva 3194 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (∀𝑥𝐵 (𝑥 ∈ (𝐼‘(𝑠𝑡)) ↔ 𝑥 ∈ ((𝐼𝑠) ∩ (𝐼𝑡))) ↔ ∀𝑥𝐵 ((𝑠𝑡) ∈ (𝑁𝑥) ↔ (𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥)))))
5233, 51bitrd 271 . . . . 5 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → ((𝐼‘(𝑠𝑡)) = ((𝐼𝑠) ∩ (𝐼𝑡)) ↔ ∀𝑥𝐵 ((𝑠𝑡) ∈ (𝑁𝑥) ↔ (𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥)))))
5352ralbidva 3194 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵) → (∀𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠𝑡)) = ((𝐼𝑠) ∩ (𝐼𝑡)) ↔ ∀𝑡 ∈ 𝒫 𝐵𝑥𝐵 ((𝑠𝑡) ∈ (𝑁𝑥) ↔ (𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥)))))
54 ralcom 3308 . . . 4 (∀𝑡 ∈ 𝒫 𝐵𝑥𝐵 ((𝑠𝑡) ∈ (𝑁𝑥) ↔ (𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥))) ↔ ∀𝑥𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) ∈ (𝑁𝑥) ↔ (𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥))))
5553, 54syl6bb 279 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵) → (∀𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠𝑡)) = ((𝐼𝑠) ∩ (𝐼𝑡)) ↔ ∀𝑥𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) ∈ (𝑁𝑥) ↔ (𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥)))))
5655ralbidva 3194 . 2 (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠𝑡)) = ((𝐼𝑠) ∩ (𝐼𝑡)) ↔ ∀𝑠 ∈ 𝒫 𝐵𝑥𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) ∈ (𝑁𝑥) ↔ (𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥)))))
57 ralcom 3308 . 2 (∀𝑠 ∈ 𝒫 𝐵𝑥𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) ∈ (𝑁𝑥) ↔ (𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥))) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) ∈ (𝑁𝑥) ↔ (𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥))))
5856, 57syl6bb 279 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 198  wa 386   = wceq 1656  wcel 2164  wral 3117  {crab 3121  Vcvv 3414  cin 3797  wss 3798  𝒫 cpw 4380   class class class wbr 4875  cmpt 4954  wf 6123  cfv 6127  (class class class)co 6910  cmpt2 6912  𝑚 cmap 8127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-1st 7433  df-2nd 7434  df-map 8129
This theorem is referenced by: (None)
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