Theorem ntrneiiso 41701

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 3907	. . . . . . . 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 1942	. . . . . . 7 ⊢ (∀𝑥(𝑠 ⊆ 𝑡 → (𝑥 ∈ (𝐼‘𝑠) → 𝑥 ∈ (𝐼‘𝑡))) ↔ (𝑠 ⊆ 𝑡 → ∀𝑥(𝑥 ∈ (𝐼‘𝑠) → 𝑥 ∈ (𝐼‘𝑡))))
4	2, 3	bitr4i 277	. . . . . 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 41686	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
11		elmapi 8637	. . . . . . . . . . . . . . . . . . 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	ffvelrnd 6962	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐼‘𝑠) ∈ 𝒫 𝐵)
15	14	elpwid 4544	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐼‘𝑠) ⊆ 𝐵)
16	15	sselda 3921	. . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . 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 1923	. . . . . . . 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 3069	. . . . . . . 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 485	. . . . . . . . . . . 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 41691	. . . . . . . . . . 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 41691	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ (𝐼‘𝑡) ↔ 𝑡 ∈ (𝑁‘𝑥)))
33	30, 32	imbi12d 345	. . . . . . . . . 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 341	. . . . . . . . 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 451	. . . . . . . . . 10 ⊢ (((𝑠 ⊆ 𝑡 ∧ 𝑠 ∈ (𝑁‘𝑥)) → 𝑡 ∈ (𝑁‘𝑥)) ↔ (𝑠 ⊆ 𝑡 → (𝑠 ∈ (𝑁‘𝑥) → 𝑡 ∈ (𝑁‘𝑥))))
36		ancomst 465	. . . . . . . . . 10 ⊢ (((𝑠 ⊆ 𝑡 ∧ 𝑠 ∈ (𝑁‘𝑥)) → 𝑡 ∈ (𝑁‘𝑥)) ↔ ((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝑡) → 𝑡 ∈ (𝑁‘𝑥)))
37	35, 36	bitr3i 276	. . . . . . . . 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 3111	. . . . . . 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 278	. . . . . 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 282	. . . . 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 3111	. . . 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 3166	. . . 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 3111	. 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 3166	. 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))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1537   = wceq 1539   ∈ wcel 2106  ∀wral 3064  {crab 3068  Vcvv 3432   ⊆ wss 3887  𝒫 cpw 4533   class class class wbr 5074   ↦ cmpt 5157  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277   ↑_m cmap 8615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-map 8617

This theorem is referenced by: (None)