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Theorem utopsnneiplem 23972

Description: The neighborhoods of a point 𝑃 for the topology induced by an uniform space 𝑈. (Contributed by Thierry Arnoux, 11-Jan-2018.)

**Hypotheses**
Ref	Expression
utoptop.1	⊢ 𝐽 = (unifTop‘𝑈)
utopsnneip.1	⊢ 𝐾 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)}
utopsnneip.2	⊢ 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))

**Assertion**
Ref	Expression
utopsnneiplem	⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) → ((nei‘𝐽)‘{𝑃}) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})))

Distinct variable groups: 𝑝,𝑎,𝐾 𝑁,𝑎,𝑝 𝑣,𝑝,𝑃 𝑣,𝑎,𝑈,𝑝 𝑋,𝑎,𝑝,𝑣 Allowed substitution hints: 𝑃(𝑎) 𝐽(𝑣,𝑝,𝑎) 𝐾(𝑣) 𝑁(𝑣)

Proof of Theorem utopsnneiplem Dummy variables 𝑏 𝑞 𝑢 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		utoptop.1	. . . . . . . 8 ⊢ 𝐽 = (unifTop‘𝑈)
2		utopval 23957	. . . . . . . 8 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝 ∈ 𝑎 ∃𝑤 ∈ 𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎})
3	1, 2	eqtrid 2782	. . . . . . 7 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝐽 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝 ∈ 𝑎 ∃𝑤 ∈ 𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎})
4		simpll 763	. . . . . . . . . . 11 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝 ∈ 𝑎) → 𝑈 ∈ (UnifOn‘𝑋))
5		simpr 483	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) → 𝑎 ∈ 𝒫 𝑋)
6	5	elpwid 4610	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) → 𝑎 ⊆ 𝑋)
7	6	sselda 3981	. . . . . . . . . . 11 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝 ∈ 𝑎) → 𝑝 ∈ 𝑋)
8		simpr 483	. . . . . . . . . . . . . 14 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → 𝑝 ∈ 𝑋)
9		mptexg 7224	. . . . . . . . . . . . . . . 16 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) ∈ V)
10		rnexg 7897	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) ∈ V → ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) ∈ V)
11	9, 10	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) ∈ V)
12	11	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) ∈ V)
13		utopsnneip.2	. . . . . . . . . . . . . . 15 ⊢ 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))
14	13	fvmpt2 7008	. . . . . . . . . . . . . 14 ⊢ ((𝑝 ∈ 𝑋 ∧ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) ∈ V) → (𝑁‘𝑝) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))
15	8, 12, 14	syl2anc 582	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → (𝑁‘𝑝) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))
16	15	eleq2d 2817	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → (𝑎 ∈ (𝑁‘𝑝) ↔ 𝑎 ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝}))))
17		eqid 2730	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) = (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝}))
18	17	elrnmpt 5954	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ V → (𝑎 ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) ↔ ∃𝑣 ∈ 𝑈 𝑎 = (𝑣 “ {𝑝})))
19	18	elv 3478	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) ↔ ∃𝑣 ∈ 𝑈 𝑎 = (𝑣 “ {𝑝}))
20	16, 19	bitrdi 286	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → (𝑎 ∈ (𝑁‘𝑝) ↔ ∃𝑣 ∈ 𝑈 𝑎 = (𝑣 “ {𝑝})))
21	4, 7, 20	syl2anc 582	. . . . . . . . . 10 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝 ∈ 𝑎) → (𝑎 ∈ (𝑁‘𝑝) ↔ ∃𝑣 ∈ 𝑈 𝑎 = (𝑣 “ {𝑝})))
22		nfv 1915	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑣((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝 ∈ 𝑎)
23		nfre1 3280	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑣∃𝑣 ∈ 𝑈 𝑎 = (𝑣 “ {𝑝})
24	22, 23	nfan 1900	. . . . . . . . . . . 12 ⊢ Ⅎ𝑣(((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝 ∈ 𝑎) ∧ ∃𝑣 ∈ 𝑈 𝑎 = (𝑣 “ {𝑝}))
25		simplr 765	. . . . . . . . . . . . 13 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝 ∈ 𝑎) ∧ ∃𝑣 ∈ 𝑈 𝑎 = (𝑣 “ {𝑝})) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑝})) → 𝑣 ∈ 𝑈)
26		eqimss2 4040	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝑣 “ {𝑝}) → (𝑣 “ {𝑝}) ⊆ 𝑎)
27	26	adantl 480	. . . . . . . . . . . . 13 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝 ∈ 𝑎) ∧ ∃𝑣 ∈ 𝑈 𝑎 = (𝑣 “ {𝑝})) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑝})) → (𝑣 “ {𝑝}) ⊆ 𝑎)
28		imaeq1 6053	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑣 → (𝑤 “ {𝑝}) = (𝑣 “ {𝑝}))
29	28	sseq1d 4012	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑣 → ((𝑤 “ {𝑝}) ⊆ 𝑎 ↔ (𝑣 “ {𝑝}) ⊆ 𝑎))
30	29	rspcev 3611	. . . . . . . . . . . . 13 ⊢ ((𝑣 ∈ 𝑈 ∧ (𝑣 “ {𝑝}) ⊆ 𝑎) → ∃𝑤 ∈ 𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎)
31	25, 27, 30	syl2anc 582	. . . . . . . . . . . 12 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝 ∈ 𝑎) ∧ ∃𝑣 ∈ 𝑈 𝑎 = (𝑣 “ {𝑝})) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑝})) → ∃𝑤 ∈ 𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎)
32		simpr 483	. . . . . . . . . . . 12 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝 ∈ 𝑎) ∧ ∃𝑣 ∈ 𝑈 𝑎 = (𝑣 “ {𝑝})) → ∃𝑣 ∈ 𝑈 𝑎 = (𝑣 “ {𝑝}))
33	24, 31, 32	r19.29af 3263	. . . . . . . . . . 11 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝 ∈ 𝑎) ∧ ∃𝑣 ∈ 𝑈 𝑎 = (𝑣 “ {𝑝})) → ∃𝑤 ∈ 𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎)
34	4	ad2antrr 722	. . . . . . . . . . . . . . 15 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝 ∈ 𝑎) ∧ 𝑤 ∈ 𝑈) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎) → 𝑈 ∈ (UnifOn‘𝑋))
35	7	ad2antrr 722	. . . . . . . . . . . . . . 15 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝 ∈ 𝑎) ∧ 𝑤 ∈ 𝑈) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎) → 𝑝 ∈ 𝑋)
36	34, 35	jca 510	. . . . . . . . . . . . . 14 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝 ∈ 𝑎) ∧ 𝑤 ∈ 𝑈) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎) → (𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋))
37		simpr 483	. . . . . . . . . . . . . 14 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝 ∈ 𝑎) ∧ 𝑤 ∈ 𝑈) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎) → (𝑤 “ {𝑝}) ⊆ 𝑎)
38	6	ad3antrrr 726	. . . . . . . . . . . . . 14 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝 ∈ 𝑎) ∧ 𝑤 ∈ 𝑈) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎) → 𝑎 ⊆ 𝑋)
39		simplr 765	. . . . . . . . . . . . . . 15 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝 ∈ 𝑎) ∧ 𝑤 ∈ 𝑈) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎) → 𝑤 ∈ 𝑈)
40		eqid 2730	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 “ {𝑝}) = (𝑤 “ {𝑝})
41		imaeq1 6053	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 = 𝑤 → (𝑢 “ {𝑝}) = (𝑤 “ {𝑝}))
42	41	rspceeqv 3632	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 ∈ 𝑈 ∧ (𝑤 “ {𝑝}) = (𝑤 “ {𝑝})) → ∃𝑢 ∈ 𝑈 (𝑤 “ {𝑝}) = (𝑢 “ {𝑝}))
43	40, 42	mpan2 687	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ 𝑈 → ∃𝑢 ∈ 𝑈 (𝑤 “ {𝑝}) = (𝑢 “ {𝑝}))
44	43	adantl 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑤 ∈ 𝑈) → ∃𝑢 ∈ 𝑈 (𝑤 “ {𝑝}) = (𝑢 “ {𝑝}))
45		vex 3476	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑤 ∈ V
46	45	imaex 7909	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 “ {𝑝}) ∈ V
47	13	ustuqtoplem 23964	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ (𝑤 “ {𝑝}) ∈ V) → ((𝑤 “ {𝑝}) ∈ (𝑁‘𝑝) ↔ ∃𝑢 ∈ 𝑈 (𝑤 “ {𝑝}) = (𝑢 “ {𝑝})))
48	46, 47	mpan2 687	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → ((𝑤 “ {𝑝}) ∈ (𝑁‘𝑝) ↔ ∃𝑢 ∈ 𝑈 (𝑤 “ {𝑝}) = (𝑢 “ {𝑝})))
49	48	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑤 ∈ 𝑈) → ((𝑤 “ {𝑝}) ∈ (𝑁‘𝑝) ↔ ∃𝑢 ∈ 𝑈 (𝑤 “ {𝑝}) = (𝑢 “ {𝑝})))
50	44, 49	mpbird 256	. . . . . . . . . . . . . . 15 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑤 ∈ 𝑈) → (𝑤 “ {𝑝}) ∈ (𝑁‘𝑝))
51	34, 35, 39, 50	syl21anc 834	. . . . . . . . . . . . . 14 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝 ∈ 𝑎) ∧ 𝑤 ∈ 𝑈) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎) → (𝑤 “ {𝑝}) ∈ (𝑁‘𝑝))
52		sseq1 4006	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = (𝑤 “ {𝑝}) → (𝑏 ⊆ 𝑎 ↔ (𝑤 “ {𝑝}) ⊆ 𝑎))
53	52	3anbi2d 1439	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = (𝑤 “ {𝑝}) → (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑏 ⊆ 𝑎 ∧ 𝑎 ⊆ 𝑋) ↔ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎 ∧ 𝑎 ⊆ 𝑋)))
54		eleq1 2819	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = (𝑤 “ {𝑝}) → (𝑏 ∈ (𝑁‘𝑝) ↔ (𝑤 “ {𝑝}) ∈ (𝑁‘𝑝)))
55	53, 54	anbi12d 629	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝑤 “ {𝑝}) → ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑏 ⊆ 𝑎 ∧ 𝑎 ⊆ 𝑋) ∧ 𝑏 ∈ (𝑁‘𝑝)) ↔ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎 ∧ 𝑎 ⊆ 𝑋) ∧ (𝑤 “ {𝑝}) ∈ (𝑁‘𝑝))))
56	55	imbi1d 340	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (𝑤 “ {𝑝}) → (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑏 ⊆ 𝑎 ∧ 𝑎 ⊆ 𝑋) ∧ 𝑏 ∈ (𝑁‘𝑝)) → 𝑎 ∈ (𝑁‘𝑝)) ↔ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎 ∧ 𝑎 ⊆ 𝑋) ∧ (𝑤 “ {𝑝}) ∈ (𝑁‘𝑝)) → 𝑎 ∈ (𝑁‘𝑝))))
57	13	ustuqtop1 23966	. . . . . . . . . . . . . . 15 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑏 ⊆ 𝑎 ∧ 𝑎 ⊆ 𝑋) ∧ 𝑏 ∈ (𝑁‘𝑝)) → 𝑎 ∈ (𝑁‘𝑝))
58	46, 56, 57	vtocl 3544	. . . . . . . . . . . . . 14 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎 ∧ 𝑎 ⊆ 𝑋) ∧ (𝑤 “ {𝑝}) ∈ (𝑁‘𝑝)) → 𝑎 ∈ (𝑁‘𝑝))
59	36, 37, 38, 51, 58	syl31anc 1371	. . . . . . . . . . . . 13 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝 ∈ 𝑎) ∧ 𝑤 ∈ 𝑈) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎) → 𝑎 ∈ (𝑁‘𝑝))
60	36, 20	syl 17	. . . . . . . . . . . . 13 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝 ∈ 𝑎) ∧ 𝑤 ∈ 𝑈) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎) → (𝑎 ∈ (𝑁‘𝑝) ↔ ∃𝑣 ∈ 𝑈 𝑎 = (𝑣 “ {𝑝})))
61	59, 60	mpbid 231	. . . . . . . . . . . 12 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝 ∈ 𝑎) ∧ 𝑤 ∈ 𝑈) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎) → ∃𝑣 ∈ 𝑈 𝑎 = (𝑣 “ {𝑝}))
62	61	r19.29an 3156	. . . . . . . . . . 11 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝 ∈ 𝑎) ∧ ∃𝑤 ∈ 𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎) → ∃𝑣 ∈ 𝑈 𝑎 = (𝑣 “ {𝑝}))
63	33, 62	impbida 797	. . . . . . . . . 10 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝 ∈ 𝑎) → (∃𝑣 ∈ 𝑈 𝑎 = (𝑣 “ {𝑝}) ↔ ∃𝑤 ∈ 𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎))
64	21, 63	bitrd 278	. . . . . . . . 9 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝 ∈ 𝑎) → (𝑎 ∈ (𝑁‘𝑝) ↔ ∃𝑤 ∈ 𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎))
65	64	ralbidva 3173	. . . . . . . 8 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) → (∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝) ↔ ∀𝑝 ∈ 𝑎 ∃𝑤 ∈ 𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎))
66	65	rabbidva 3437	. . . . . . 7 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)} = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝 ∈ 𝑎 ∃𝑤 ∈ 𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎})
67	3, 66	eqtr4d 2773	. . . . . 6 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝐽 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)})
68		utopsnneip.1	. . . . . 6 ⊢ 𝐾 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)}
69	67, 68	eqtr4di 2788	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝐽 = 𝐾)
70	69	fveq2d 6894	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (nei‘𝐽) = (nei‘𝐾))
71	70	fveq1d 6892	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ((nei‘𝐽)‘{𝑃}) = ((nei‘𝐾)‘{𝑃}))
72	71	adantr 479	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) → ((nei‘𝐽)‘{𝑃}) = ((nei‘𝐾)‘{𝑃}))
73	13	ustuqtop0 23965	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑁:𝑋⟶𝒫 𝒫 𝑋)
74	13	ustuqtop1 23966	. . . . 5 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → 𝑏 ∈ (𝑁‘𝑝))
75	13	ustuqtop2 23967	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → (fi‘(𝑁‘𝑝)) ⊆ (𝑁‘𝑝))
76	13	ustuqtop3 23968	. . . . 5 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → 𝑝 ∈ 𝑎)
77	13	ustuqtop4 23969	. . . . 5 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → ∃𝑏 ∈ (𝑁‘𝑝)∀𝑞 ∈ 𝑏 𝑎 ∈ (𝑁‘𝑞))
78	13	ustuqtop5 23970	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → 𝑋 ∈ (𝑁‘𝑝))
79	68, 73, 74, 75, 76, 77, 78	neiptopnei 22856	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝐾)‘{𝑝})))
80	79	adantr 479	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) → 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝐾)‘{𝑝})))
81		simpr 483	. . . . 5 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑝 = 𝑃) → 𝑝 = 𝑃)
82	81	sneqd 4639	. . . 4 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑝 = 𝑃) → {𝑝} = {𝑃})
83	82	fveq2d 6894	. . 3 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑝 = 𝑃) → ((nei‘𝐾)‘{𝑝}) = ((nei‘𝐾)‘{𝑃}))
84		simpr 483	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) → 𝑃 ∈ 𝑋)
85		fvexd 6905	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) → ((nei‘𝐾)‘{𝑃}) ∈ V)
86	80, 83, 84, 85	fvmptd 7004	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) → (𝑁‘𝑃) = ((nei‘𝐾)‘{𝑃}))
87		mptexg 7224	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
88		rnexg 7897	. . . . 5 ⊢ ((𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ∈ V → ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
89	87, 88	syl 17	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
90	89	adantr 479	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) → ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
91		nfv 1915	. . . . . . . 8 ⊢ Ⅎ𝑣 𝑃 ∈ 𝑋
92		nfmpt1 5255	. . . . . . . . . 10 ⊢ Ⅎ𝑣(𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃}))
93	92	nfrn 5950	. . . . . . . . 9 ⊢ Ⅎ𝑣ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃}))
94	93	nfel1 2917	. . . . . . . 8 ⊢ Ⅎ𝑣ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ∈ V
95	91, 94	nfan 1900	. . . . . . 7 ⊢ Ⅎ𝑣(𝑃 ∈ 𝑋 ∧ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
96		nfv 1915	. . . . . . 7 ⊢ Ⅎ𝑣 𝑝 = 𝑃
97	95, 96	nfan 1900	. . . . . 6 ⊢ Ⅎ𝑣((𝑃 ∈ 𝑋 ∧ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) ∧ 𝑝 = 𝑃)
98		simpr2 1193	. . . . . . . . 9 ⊢ ((𝑃 ∈ 𝑋 ∧ (ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ∈ V ∧ 𝑝 = 𝑃 ∧ 𝑣 ∈ 𝑈)) → 𝑝 = 𝑃)
99	98	sneqd 4639	. . . . . . . 8 ⊢ ((𝑃 ∈ 𝑋 ∧ (ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ∈ V ∧ 𝑝 = 𝑃 ∧ 𝑣 ∈ 𝑈)) → {𝑝} = {𝑃})
100	99	imaeq2d 6058	. . . . . . 7 ⊢ ((𝑃 ∈ 𝑋 ∧ (ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ∈ V ∧ 𝑝 = 𝑃 ∧ 𝑣 ∈ 𝑈)) → (𝑣 “ {𝑝}) = (𝑣 “ {𝑃}))
101	100	3anassrs 1358	. . . . . 6 ⊢ ((((𝑃 ∈ 𝑋 ∧ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) ∧ 𝑝 = 𝑃) ∧ 𝑣 ∈ 𝑈) → (𝑣 “ {𝑝}) = (𝑣 “ {𝑃}))
102	97, 101	mpteq2da 5245	. . . . 5 ⊢ (((𝑃 ∈ 𝑋 ∧ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) ∧ 𝑝 = 𝑃) → (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) = (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})))
103	102	rneqd 5936	. . . 4 ⊢ (((𝑃 ∈ 𝑋 ∧ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) ∧ 𝑝 = 𝑃) → ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})))
104		simpl 481	. . . 4 ⊢ ((𝑃 ∈ 𝑋 ∧ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) → 𝑃 ∈ 𝑋)
105		simpr 483	. . . 4 ⊢ ((𝑃 ∈ 𝑋 ∧ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) → ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
106	13, 103, 104, 105	fvmptd2 7005	. . 3 ⊢ ((𝑃 ∈ 𝑋 ∧ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) → (𝑁‘𝑃) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})))
107	84, 90, 106	syl2anc 582	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) → (𝑁‘𝑃) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})))
108	72, 86, 107	3eqtr2d 2776	1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) → ((nei‘𝐽)‘{𝑃}) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1085 = wceq 1539 ∈ wcel 2104 ∀wral 3059 ∃wrex 3068 {crab 3430 Vcvv 3472 ⊆ wss 3947 𝒫 cpw 4601 {csn 4627 ↦ cmpt 5230 ran crn 5676 “ cima 5678 ‘cfv 6542 neicnei 22821 UnifOncust 23924 unifTopcutop 23955

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-om 7858 df-1o 8468 df-er 8705 df-en 8942 df-fin 8945 df-fi 9408 df-top 22616 df-nei 22822 df-ust 23925 df-utop 23956

This theorem is referenced by: utopsnneip 23973

W3C validator