ntrneiiso - Mathbox for Richard Penner

	Mathbox for Richard Penner		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > ntrneiiso		Structured version Visualization version GIF version

Theorem ntrneiiso 43552

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 3969	. . . . . . . 8 ⊢ ((𝐼‘𝑠) ⊆ (𝐼‘𝑡) ↔ ∀𝑥(𝑥 ∈ (𝐼‘𝑠) → 𝑥 ∈ (𝐼‘𝑡)))
2	1	imbi2i 335	. . . . . . 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 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 765	. . . . . . . . . . . . . . . . . . 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 43537	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵))
11		elmapi 8874	. . . . . . . . . . . . . . . . . . 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 767	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
14	12, 13	ffvelcdmd 7100	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐼‘𝑠) ∈ 𝒫 𝐵)
15	14	elpwid 4615	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐼‘𝑠) ⊆ 𝐵)
16	15	sselda 3982	. . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . 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 3059	. . . . . . . 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 288	. . . . . . 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 728	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝐼𝐹𝑁)
28		simpr 483	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
29		simpllr 774	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑠 ∈ 𝒫 𝐵)
30	7, 8, 27, 28, 29	ntrneiel 43542	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ (𝐼‘𝑠) ↔ 𝑠 ∈ (𝑁‘𝑥)))
31		simplr 767	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑡 ∈ 𝒫 𝐵)
32	7, 8, 27, 28, 31	ntrneiel 43542	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ (𝐼‘𝑡) ↔ 𝑡 ∈ (𝑁‘𝑥)))
33	30, 32	imbi12d 343	. . . . . . . . . 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 339	. . . . . . . . 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 449	. . . . . . . . . 10 ⊢ (((𝑠 ⊆ 𝑡 ∧ 𝑠 ∈ (𝑁‘𝑥)) → 𝑡 ∈ (𝑁‘𝑥)) ↔ (𝑠 ⊆ 𝑡 → (𝑠 ∈ (𝑁‘𝑥) → 𝑡 ∈ (𝑁‘𝑥))))
36		ancomst 463	. . . . . . . . . 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 286	. . . . . . . 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 3173	. . . . . . 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 3173	. . . 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 3284	. . . 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 286	. . 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 3173	. 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 3284	. 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 286	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 394 ∀wal 1531 = wceq 1533 ∈ wcel 2098 ∀wral 3058 {crab 3430 Vcvv 3473 ⊆ wss 3949 𝒫 cpw 4606 class class class wbr 5152 ↦ cmpt 5235 ⟶wf 6549 ‘cfv 6553 (class class class)co 7426 ∈ cmpo 7428 ↑_m cmap 8851

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 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746

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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 7999 df-2nd 8000 df-map 8853

This theorem is referenced by: (None)

W3C validator