ntrneiiso - Mathbox for Richard Penner

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Theorem ntrneiiso 43400

Description: If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then conditions equal to claiming that the interior function is isotonic hold equally. (Contributed by RP, 3-Jun-2021.)

**Hypotheses**
Ref	Expression
ntrnei.o	⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑_m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))
ntrnei.f	⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)
ntrnei.r	⊢ (𝜑 → 𝐼𝐹𝑁)

**Assertion**
Ref	Expression
ntrneiiso	⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝑠 ⊆ 𝑡 → (𝐼‘𝑠) ⊆ (𝐼‘𝑡)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝑡) → 𝑡 ∈ (𝑁‘𝑥))))

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 ntrneiiso**
Step	Hyp	Ref	Expression
1		dfss2 3963	. . . . . . . 8 ⊢ ((𝐼‘𝑠) ⊆ (𝐼‘𝑡) ↔ ∀𝑥(𝑥 ∈ (𝐼‘𝑠) → 𝑥 ∈ (𝐼‘𝑡)))
2	1	imbi2i 336	. . . . . . 7 ⊢ ((𝑠 ⊆ 𝑡 → (𝐼‘𝑠) ⊆ (𝐼‘𝑡)) ↔ (𝑠 ⊆ 𝑡 → ∀𝑥(𝑥 ∈ (𝐼‘𝑠) → 𝑥 ∈ (𝐼‘𝑡))))
3		19.21v 1934	. . . . . . 7 ⊢ (∀𝑥(𝑠 ⊆ 𝑡 → (𝑥 ∈ (𝐼‘𝑠) → 𝑥 ∈ (𝐼‘𝑡))) ↔ (𝑠 ⊆ 𝑡 → ∀𝑥(𝑥 ∈ (𝐼‘𝑠) → 𝑥 ∈ (𝐼‘𝑡))))
4	2, 3	bitr4i 278	. . . . . 6 ⊢ ((𝑠 ⊆ 𝑡 → (𝐼‘𝑠) ⊆ (𝐼‘𝑡)) ↔ ∀𝑥(𝑠 ⊆ 𝑡 → (𝑥 ∈ (𝐼‘𝑠) → 𝑥 ∈ (𝐼‘𝑡))))
5		ax-1 6	. . . . . . . . . 10 ⊢ ((𝑠 ⊆ 𝑡 → (𝑥 ∈ (𝐼‘𝑠) → 𝑥 ∈ (𝐼‘𝑡))) → (𝑥 ∈ 𝐵 → (𝑠 ⊆ 𝑡 → (𝑥 ∈ (𝐼‘𝑠) → 𝑥 ∈ (𝐼‘𝑡)))))
6		simpll 764	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝜑)
7		ntrnei.o	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑_m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))
8		ntrnei.f	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)
9		ntrnei.r	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐼𝐹𝑁)
10	7, 8, 9	ntrneiiex 43385	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵))
11		elmapi 8842	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
12	6, 10, 11	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
13		simplr 766	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
14	12, 13	ffvelcdmd 7080	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐼‘𝑠) ∈ 𝒫 𝐵)
15	14	elpwid 4606	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐼‘𝑠) ⊆ 𝐵)
16	15	sselda 3977	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ (𝐼‘𝑠)) → 𝑥 ∈ 𝐵)
17	16	pm2.24d 151	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ (𝐼‘𝑠)) → (¬ 𝑥 ∈ 𝐵 → 𝑥 ∈ (𝐼‘𝑡)))
18	17	ex 412	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝑥 ∈ (𝐼‘𝑠) → (¬ 𝑥 ∈ 𝐵 → 𝑥 ∈ (𝐼‘𝑡))))
19	18	com23 86	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (¬ 𝑥 ∈ 𝐵 → (𝑥 ∈ (𝐼‘𝑠) → 𝑥 ∈ (𝐼‘𝑡))))
20	19	a1dd 50	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (¬ 𝑥 ∈ 𝐵 → (𝑠 ⊆ 𝑡 → (𝑥 ∈ (𝐼‘𝑠) → 𝑥 ∈ (𝐼‘𝑡)))))
21		idd 24	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → ((𝑠 ⊆ 𝑡 → (𝑥 ∈ (𝐼‘𝑠) → 𝑥 ∈ (𝐼‘𝑡))) → (𝑠 ⊆ 𝑡 → (𝑥 ∈ (𝐼‘𝑠) → 𝑥 ∈ (𝐼‘𝑡)))))
22	20, 21	jad 187	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → ((𝑥 ∈ 𝐵 → (𝑠 ⊆ 𝑡 → (𝑥 ∈ (𝐼‘𝑠) → 𝑥 ∈ (𝐼‘𝑡)))) → (𝑠 ⊆ 𝑡 → (𝑥 ∈ (𝐼‘𝑠) → 𝑥 ∈ (𝐼‘𝑡)))))
23	5, 22	impbid2 225	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → ((𝑠 ⊆ 𝑡 → (𝑥 ∈ (𝐼‘𝑠) → 𝑥 ∈ (𝐼‘𝑡))) ↔ (𝑥 ∈ 𝐵 → (𝑠 ⊆ 𝑡 → (𝑥 ∈ (𝐼‘𝑠) → 𝑥 ∈ (𝐼‘𝑡))))))
24	23	albidv 1915	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (∀𝑥(𝑠 ⊆ 𝑡 → (𝑥 ∈ (𝐼‘𝑠) → 𝑥 ∈ (𝐼‘𝑡))) ↔ ∀𝑥(𝑥 ∈ 𝐵 → (𝑠 ⊆ 𝑡 → (𝑥 ∈ (𝐼‘𝑠) → 𝑥 ∈ (𝐼‘𝑡))))))
25		df-ral 3056	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐵 (𝑠 ⊆ 𝑡 → (𝑥 ∈ (𝐼‘𝑠) → 𝑥 ∈ (𝐼‘𝑡))) ↔ ∀𝑥(𝑥 ∈ 𝐵 → (𝑠 ⊆ 𝑡 → (𝑥 ∈ (𝐼‘𝑠) → 𝑥 ∈ (𝐼‘𝑡)))))
26	24, 25	bitr4di 289	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (∀𝑥(𝑠 ⊆ 𝑡 → (𝑥 ∈ (𝐼‘𝑠) → 𝑥 ∈ (𝐼‘𝑡))) ↔ ∀𝑥 ∈ 𝐵 (𝑠 ⊆ 𝑡 → (𝑥 ∈ (𝐼‘𝑠) → 𝑥 ∈ (𝐼‘𝑡)))))
27	9	ad3antrrr 727	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝐼𝐹𝑁)
28		simpr 484	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
29		simpllr 773	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑠 ∈ 𝒫 𝐵)
30	7, 8, 27, 28, 29	ntrneiel 43390	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ (𝐼‘𝑠) ↔ 𝑠 ∈ (𝑁‘𝑥)))
31		simplr 766	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑡 ∈ 𝒫 𝐵)
32	7, 8, 27, 28, 31	ntrneiel 43390	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ (𝐼‘𝑡) ↔ 𝑡 ∈ (𝑁‘𝑥)))
33	30, 32	imbi12d 344	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → ((𝑥 ∈ (𝐼‘𝑠) → 𝑥 ∈ (𝐼‘𝑡)) ↔ (𝑠 ∈ (𝑁‘𝑥) → 𝑡 ∈ (𝑁‘𝑥))))
34	33	imbi2d 340	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → ((𝑠 ⊆ 𝑡 → (𝑥 ∈ (𝐼‘𝑠) → 𝑥 ∈ (𝐼‘𝑡))) ↔ (𝑠 ⊆ 𝑡 → (𝑠 ∈ (𝑁‘𝑥) → 𝑡 ∈ (𝑁‘𝑥)))))
35		impexp 450	. . . . . . . . . 10 ⊢ (((𝑠 ⊆ 𝑡 ∧ 𝑠 ∈ (𝑁‘𝑥)) → 𝑡 ∈ (𝑁‘𝑥)) ↔ (𝑠 ⊆ 𝑡 → (𝑠 ∈ (𝑁‘𝑥) → 𝑡 ∈ (𝑁‘𝑥))))
36		ancomst 464	. . . . . . . . . 10 ⊢ (((𝑠 ⊆ 𝑡 ∧ 𝑠 ∈ (𝑁‘𝑥)) → 𝑡 ∈ (𝑁‘𝑥)) ↔ ((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝑡) → 𝑡 ∈ (𝑁‘𝑥)))
37	35, 36	bitr3i 277	. . . . . . . . 9 ⊢ ((𝑠 ⊆ 𝑡 → (𝑠 ∈ (𝑁‘𝑥) → 𝑡 ∈ (𝑁‘𝑥))) ↔ ((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝑡) → 𝑡 ∈ (𝑁‘𝑥)))
38	34, 37	bitrdi 287	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → ((𝑠 ⊆ 𝑡 → (𝑥 ∈ (𝐼‘𝑠) → 𝑥 ∈ (𝐼‘𝑡))) ↔ ((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝑡) → 𝑡 ∈ (𝑁‘𝑥))))
39	38	ralbidva 3169	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (∀𝑥 ∈ 𝐵 (𝑠 ⊆ 𝑡 → (𝑥 ∈ (𝐼‘𝑠) → 𝑥 ∈ (𝐼‘𝑡))) ↔ ∀𝑥 ∈ 𝐵 ((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝑡) → 𝑡 ∈ (𝑁‘𝑥))))
40	26, 39	bitrd 279	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (∀𝑥(𝑠 ⊆ 𝑡 → (𝑥 ∈ (𝐼‘𝑠) → 𝑥 ∈ (𝐼‘𝑡))) ↔ ∀𝑥 ∈ 𝐵 ((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝑡) → 𝑡 ∈ (𝑁‘𝑥))))
41	4, 40	bitrid 283	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → ((𝑠 ⊆ 𝑡 → (𝐼‘𝑠) ⊆ (𝐼‘𝑡)) ↔ ∀𝑥 ∈ 𝐵 ((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝑡) → 𝑡 ∈ (𝑁‘𝑥))))
42	41	ralbidva 3169	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (∀𝑡 ∈ 𝒫 𝐵(𝑠 ⊆ 𝑡 → (𝐼‘𝑠) ⊆ (𝐼‘𝑡)) ↔ ∀𝑡 ∈ 𝒫 𝐵∀𝑥 ∈ 𝐵 ((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝑡) → 𝑡 ∈ (𝑁‘𝑥))))
43		ralcom 3280	. . . 4 ⊢ (∀𝑡 ∈ 𝒫 𝐵∀𝑥 ∈ 𝐵 ((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝑡) → 𝑡 ∈ (𝑁‘𝑥)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝑡) → 𝑡 ∈ (𝑁‘𝑥)))
44	42, 43	bitrdi 287	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (∀𝑡 ∈ 𝒫 𝐵(𝑠 ⊆ 𝑡 → (𝐼‘𝑠) ⊆ (𝐼‘𝑡)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝑡) → 𝑡 ∈ (𝑁‘𝑥))))
45	44	ralbidva 3169	. 2 ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝑠 ⊆ 𝑡 → (𝐼‘𝑠) ⊆ (𝐼‘𝑡)) ↔ ∀𝑠 ∈ 𝒫 𝐵∀𝑥 ∈ 𝐵 ∀𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝑡) → 𝑡 ∈ (𝑁‘𝑥))))
46		ralcom 3280	. 2 ⊢ (∀𝑠 ∈ 𝒫 𝐵∀𝑥 ∈ 𝐵 ∀𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝑡) → 𝑡 ∈ (𝑁‘𝑥)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝑡) → 𝑡 ∈ (𝑁‘𝑥)))
47	45, 46	bitrdi 287	1 ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝑠 ⊆ 𝑡 → (𝐼‘𝑠) ⊆ (𝐼‘𝑡)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝑡) → 𝑡 ∈ (𝑁‘𝑥))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∀wal 1531 = wceq 1533 ∈ wcel 2098 ∀wral 3055 {crab 3426 Vcvv 3468 ⊆ wss 3943 𝒫 cpw 4597 class class class wbr 5141 ↦ cmpt 5224 ⟶wf 6532 ‘cfv 6536 (class class class)co 7404 ∈ cmpo 7406 ↑_m cmap 8819

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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7971 df-2nd 7972 df-map 8821

This theorem is referenced by: (None)

W3C validator