Theorem ntrneiiso 44918

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		df-ss 3919	. . . . . . . 8 ⊢ ((𝐼‘𝑠) ⊆ (𝐼‘𝑡) ↔ ∀𝑥(𝑥 ∈ (𝐼‘𝑠) → 𝑥 ∈ (𝐼‘𝑡)))
2	1	imbi2i 339	. . . . . . 7 ⊢ ((𝑠 ⊆ 𝑡 → (𝐼‘𝑠) ⊆ (𝐼‘𝑡)) ↔ (𝑠 ⊆ 𝑡 → ∀𝑥(𝑥 ∈ (𝐼‘𝑠) → 𝑥 ∈ (𝐼‘𝑡))))
3		19.21v 1972	. . . . . . 7 ⊢ (∀𝑥(𝑠 ⊆ 𝑡 → (𝑥 ∈ (𝐼‘𝑠) → 𝑥 ∈ (𝐼‘𝑡))) ↔ (𝑠 ⊆ 𝑡 → ∀𝑥(𝑥 ∈ (𝐼‘𝑠) → 𝑥 ∈ (𝐼‘𝑡))))
4	2, 3	bitr4i 281	. . . . . 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 779	. . . . . . . . . . . . . . . . . . 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 44903	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
11		elmapi 8851	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
12	6, 10, 11	3syl 19	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
13		simplr 781	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
14	12, 13	ffvelcdmd 7081	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐼‘𝑠) ∈ 𝒫 𝐵)
15	14	elpwid 4569	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐼‘𝑠) ⊆ 𝐵)
16	15	sselda 3934	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ (𝐼‘𝑠)) → 𝑥 ∈ 𝐵)
17	16	pm2.24d 152	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ (𝐼‘𝑠)) → (¬ 𝑥 ∈ 𝐵 → 𝑥 ∈ (𝐼‘𝑡)))
18	17	ex 418	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝑥 ∈ (𝐼‘𝑠) → (¬ 𝑥 ∈ 𝐵 → 𝑥 ∈ (𝐼‘𝑡))))
19	18	com23 87	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (¬ 𝑥 ∈ 𝐵 → (𝑥 ∈ (𝐼‘𝑠) → 𝑥 ∈ (𝐼‘𝑡))))
20	19	a1dd 51	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (¬ 𝑥 ∈ 𝐵 → (𝑠 ⊆ 𝑡 → (𝑥 ∈ (𝐼‘𝑠) → 𝑥 ∈ (𝐼‘𝑡)))))
21		idd 25	. . . . . . . . . . 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 189	. . . . . . . . . 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 229	. . . . . . . . 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 1953	. . . . . . . 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 3079	. . . . . . . 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 292	. . . . . . 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 743	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝐼𝐹𝑁)
28		simpr 490	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
29		simpllr 788	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑠 ∈ 𝒫 𝐵)
30	7, 8, 27, 28, 29	ntrneiel 44908	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ (𝐼‘𝑠) ↔ 𝑠 ∈ (𝑁‘𝑥)))
31		simplr 781	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑡 ∈ 𝒫 𝐵)
32	7, 8, 27, 28, 31	ntrneiel 44908	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ (𝐼‘𝑡) ↔ 𝑡 ∈ (𝑁‘𝑥)))
33	30, 32	imbi12d 347	. . . . . . . . . 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 343	. . . . . . . . 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 456	. . . . . . . . . 10 ⊢ (((𝑠 ⊆ 𝑡 ∧ 𝑠 ∈ (𝑁‘𝑥)) → 𝑡 ∈ (𝑁‘𝑥)) ↔ (𝑠 ⊆ 𝑡 → (𝑠 ∈ (𝑁‘𝑥) → 𝑡 ∈ (𝑁‘𝑥))))
36		ancomst 470	. . . . . . . . . 10 ⊢ (((𝑠 ⊆ 𝑡 ∧ 𝑠 ∈ (𝑁‘𝑥)) → 𝑡 ∈ (𝑁‘𝑥)) ↔ ((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝑡) → 𝑡 ∈ (𝑁‘𝑥)))
37	35, 36	bitr3i 280	. . . . . . . . 9 ⊢ ((𝑠 ⊆ 𝑡 → (𝑠 ∈ (𝑁‘𝑥) → 𝑡 ∈ (𝑁‘𝑥))) ↔ ((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝑡) → 𝑡 ∈ (𝑁‘𝑥)))
38	34, 37	bitrdi 290	. . . . . . . 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 3185	. . . . . . 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 282	. . . . . 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 286	. . . . 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 3185	. . . 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 3292	. . . 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 290	. . 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 3185	. 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 3292	. 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 290	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))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2145  ∀wral 3078  {crab 3414  Vcvv 3453   ⊆ wss 3902  𝒫 cpw 4560   class class class wbr 5107   ↦ cmpt 5190  ⟶wf 6533  ‘cfv 6537  (class class class)co 7416   ∈ cmpo 7418   ↑_m cmap 8829

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7989  df-2nd 7990  df-map 8831

This theorem is used by: (None)