Theorem ntrneicls00 44785

Description: If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then conditions equal to claiming that the closure of the empty set is the empty set hold equally. (Contributed by RP, 2-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
ntrneicls00	⊢ (𝜑 → ((𝐼‘𝐵) = 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐵 ∈ (𝑁‘𝑥)))

Distinct variable groups:   𝐵,𝑖,𝑗,𝑘,𝑙,𝑚,𝑥   𝑘,𝐼,𝑙,𝑚,𝑥   𝜑,𝑖,𝑗,𝑘,𝑙,𝑥

Allowed substitution hints:   𝜑(𝑚)   𝐹(𝑥,𝑖,𝑗,𝑘,𝑚,𝑙)   𝐼(𝑖,𝑗)   𝑁(𝑥,𝑖,𝑗,𝑘,𝑚,𝑙)   𝑂(𝑥,𝑖,𝑗,𝑘,𝑚,𝑙)

Proof of Theorem ntrneicls00

Step	Hyp	Ref	Expression
1		ntrnei.o	. . . . . . 7 ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))
2		ntrnei.f	. . . . . . 7 ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)
3		ntrnei.r	. . . . . . 7 ⊢ (𝜑 → 𝐼𝐹𝑁)
4	1, 2, 3	ntrneiiex 44772	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
5		elmapi 8845	. . . . . 6 ⊢ (𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
6	4, 5	syl 18	. . . . 5 ⊢ (𝜑 → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
7	1, 2, 3	ntrneibex 44769	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ V)
8		pwidg 4581	. . . . . 6 ⊢ (𝐵 ∈ V → 𝐵 ∈ 𝒫 𝐵)
9	7, 8	syl 18	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝒫 𝐵)
10	6, 9	ffvelcdmd 7080	. . . 4 ⊢ (𝜑 → (𝐼‘𝐵) ∈ 𝒫 𝐵)
11	10	elpwid 4570	. . 3 ⊢ (𝜑 → (𝐼‘𝐵) ⊆ 𝐵)
12		eqss 3951	. . . . 5 ⊢ ((𝐼‘𝐵) = 𝐵 ↔ ((𝐼‘𝐵) ⊆ 𝐵 ∧ 𝐵 ⊆ (𝐼‘𝐵)))
13		dfss3 3925	. . . . . 6 ⊢ (𝐵 ⊆ (𝐼‘𝐵) ↔ ∀𝑥 ∈ 𝐵 𝑥 ∈ (𝐼‘𝐵))
14	13	anbi2i 634	. . . . 5 ⊢ (((𝐼‘𝐵) ⊆ 𝐵 ∧ 𝐵 ⊆ (𝐼‘𝐵)) ↔ ((𝐼‘𝐵) ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐵 𝑥 ∈ (𝐼‘𝐵)))
15	12, 14	bitri 278	. . . 4 ⊢ ((𝐼‘𝐵) = 𝐵 ↔ ((𝐼‘𝐵) ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐵 𝑥 ∈ (𝐼‘𝐵)))
16	15	a1i 11	. . 3 ⊢ (𝜑 → ((𝐼‘𝐵) = 𝐵 ↔ ((𝐼‘𝐵) ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐵 𝑥 ∈ (𝐼‘𝐵))))
17	11, 16	mpbirand 719	. 2 ⊢ (𝜑 → ((𝐼‘𝐵) = 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝑥 ∈ (𝐼‘𝐵)))
18	3	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐼𝐹𝑁)
19		simpr 489	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
20	9	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐵 ∈ 𝒫 𝐵)
21	1, 2, 18, 19, 20	ntrneiel 44777	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ (𝐼‘𝐵) ↔ 𝐵 ∈ (𝑁‘𝑥)))
22	21	ralbidva 3184	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝑥 ∈ (𝐼‘𝐵) ↔ ∀𝑥 ∈ 𝐵 𝐵 ∈ (𝑁‘𝑥)))
23	17, 22	bitrd 282	1 ⊢ (𝜑 → ((𝐼‘𝐵) = 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐵 ∈ (𝑁‘𝑥)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1568   ∈ wcel 2141  ∀wral 3077  {crab 3414  Vcvv 3453   ⊆ wss 3904  𝒫 cpw 4561   class class class wbr 5108   ↦ cmpt 5191  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412   ↑_m cmap 8823

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7985  df-2nd 7986  df-map 8825

This theorem is referenced by: (None)