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Theorem utopsnneiplem 22250
 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 ∣ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)}
utopsnneip.2 𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
Assertion
Ref Expression
utopsnneiplem ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → ((nei‘𝐽)‘{𝑃}) = ran (𝑣𝑈 ↦ (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.
StepHypRef Expression
1 utoptop.1 . . . . . . . 8 𝐽 = (unifTop‘𝑈)
2 utopval 22235 . . . . . . . 8 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎𝑤𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎})
31, 2syl5eq 2804 . . . . . . 7 (𝑈 ∈ (UnifOn‘𝑋) → 𝐽 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎𝑤𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎})
4 simpll 807 . . . . . . . . . . 11 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) → 𝑈 ∈ (UnifOn‘𝑋))
5 simpr 479 . . . . . . . . . . . . 13 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) → 𝑎 ∈ 𝒫 𝑋)
65elpwid 4312 . . . . . . . . . . . 12 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) → 𝑎𝑋)
76sselda 3742 . . . . . . . . . . 11 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) → 𝑝𝑋)
8 simpr 479 . . . . . . . . . . . . . 14 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → 𝑝𝑋)
9 mptexg 6646 . . . . . . . . . . . . . . . 16 (𝑈 ∈ (UnifOn‘𝑋) → (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ V)
10 rnexg 7261 . . . . . . . . . . . . . . . 16 ((𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ V → ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ V)
119, 10syl 17 . . . . . . . . . . . . . . 15 (𝑈 ∈ (UnifOn‘𝑋) → ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ V)
1211adantr 472 . . . . . . . . . . . . . 14 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ V)
13 utopsnneip.2 . . . . . . . . . . . . . . 15 𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
1413fvmpt2 6451 . . . . . . . . . . . . . 14 ((𝑝𝑋 ∧ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ V) → (𝑁𝑝) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
158, 12, 14syl2anc 696 . . . . . . . . . . . . 13 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (𝑁𝑝) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
1615eleq2d 2823 . . . . . . . . . . . 12 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (𝑎 ∈ (𝑁𝑝) ↔ 𝑎 ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝}))))
17 vex 3341 . . . . . . . . . . . . 13 𝑎 ∈ V
18 eqid 2758 . . . . . . . . . . . . . 14 (𝑣𝑈 ↦ (𝑣 “ {𝑝})) = (𝑣𝑈 ↦ (𝑣 “ {𝑝}))
1918elrnmpt 5525 . . . . . . . . . . . . 13 (𝑎 ∈ V → (𝑎 ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ↔ ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝})))
2017, 19ax-mp 5 . . . . . . . . . . . 12 (𝑎 ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ↔ ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝}))
2116, 20syl6bb 276 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (𝑎 ∈ (𝑁𝑝) ↔ ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝})))
224, 7, 21syl2anc 696 . . . . . . . . . 10 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) → (𝑎 ∈ (𝑁𝑝) ↔ ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝})))
23 nfv 1990 . . . . . . . . . . . . 13 𝑣((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎)
24 nfre1 3141 . . . . . . . . . . . . 13 𝑣𝑣𝑈 𝑎 = (𝑣 “ {𝑝})
2523, 24nfan 1975 . . . . . . . . . . . 12 𝑣(((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝}))
26 simplr 809 . . . . . . . . . . . . 13 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝})) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑝})) → 𝑣𝑈)
27 eqimss2 3797 . . . . . . . . . . . . . 14 (𝑎 = (𝑣 “ {𝑝}) → (𝑣 “ {𝑝}) ⊆ 𝑎)
2827adantl 473 . . . . . . . . . . . . 13 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝})) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑝})) → (𝑣 “ {𝑝}) ⊆ 𝑎)
29 imaeq1 5617 . . . . . . . . . . . . . . 15 (𝑤 = 𝑣 → (𝑤 “ {𝑝}) = (𝑣 “ {𝑝}))
3029sseq1d 3771 . . . . . . . . . . . . . 14 (𝑤 = 𝑣 → ((𝑤 “ {𝑝}) ⊆ 𝑎 ↔ (𝑣 “ {𝑝}) ⊆ 𝑎))
3130rspcev 3447 . . . . . . . . . . . . 13 ((𝑣𝑈 ∧ (𝑣 “ {𝑝}) ⊆ 𝑎) → ∃𝑤𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎)
3226, 28, 31syl2anc 696 . . . . . . . . . . . 12 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝})) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑝})) → ∃𝑤𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎)
33 simpr 479 . . . . . . . . . . . 12 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝})) → ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝}))
3425, 32, 33r19.29af 3212 . . . . . . . . . . 11 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝})) → ∃𝑤𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎)
354ad2antrr 764 . . . . . . . . . . . . . . 15 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ 𝑤𝑈) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎) → 𝑈 ∈ (UnifOn‘𝑋))
367ad2antrr 764 . . . . . . . . . . . . . . 15 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ 𝑤𝑈) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎) → 𝑝𝑋)
3735, 36jca 555 . . . . . . . . . . . . . 14 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ 𝑤𝑈) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎) → (𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋))
38 simpr 479 . . . . . . . . . . . . . 14 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ 𝑤𝑈) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎) → (𝑤 “ {𝑝}) ⊆ 𝑎)
396ad3antrrr 768 . . . . . . . . . . . . . 14 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ 𝑤𝑈) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎) → 𝑎𝑋)
40 simplr 809 . . . . . . . . . . . . . . 15 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ 𝑤𝑈) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎) → 𝑤𝑈)
41 eqid 2758 . . . . . . . . . . . . . . . . . 18 (𝑤 “ {𝑝}) = (𝑤 “ {𝑝})
42 imaeq1 5617 . . . . . . . . . . . . . . . . . . . 20 (𝑢 = 𝑤 → (𝑢 “ {𝑝}) = (𝑤 “ {𝑝}))
4342eqeq2d 2768 . . . . . . . . . . . . . . . . . . 19 (𝑢 = 𝑤 → ((𝑤 “ {𝑝}) = (𝑢 “ {𝑝}) ↔ (𝑤 “ {𝑝}) = (𝑤 “ {𝑝})))
4443rspcev 3447 . . . . . . . . . . . . . . . . . 18 ((𝑤𝑈 ∧ (𝑤 “ {𝑝}) = (𝑤 “ {𝑝})) → ∃𝑢𝑈 (𝑤 “ {𝑝}) = (𝑢 “ {𝑝}))
4541, 44mpan2 709 . . . . . . . . . . . . . . . . 17 (𝑤𝑈 → ∃𝑢𝑈 (𝑤 “ {𝑝}) = (𝑢 “ {𝑝}))
4645adantl 473 . . . . . . . . . . . . . . . 16 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑤𝑈) → ∃𝑢𝑈 (𝑤 “ {𝑝}) = (𝑢 “ {𝑝}))
47 vex 3341 . . . . . . . . . . . . . . . . . . 19 𝑤 ∈ V
4847imaex 7267 . . . . . . . . . . . . . . . . . 18 (𝑤 “ {𝑝}) ∈ V
4913ustuqtoplem 22242 . . . . . . . . . . . . . . . . . 18 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ (𝑤 “ {𝑝}) ∈ V) → ((𝑤 “ {𝑝}) ∈ (𝑁𝑝) ↔ ∃𝑢𝑈 (𝑤 “ {𝑝}) = (𝑢 “ {𝑝})))
5048, 49mpan2 709 . . . . . . . . . . . . . . . . 17 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → ((𝑤 “ {𝑝}) ∈ (𝑁𝑝) ↔ ∃𝑢𝑈 (𝑤 “ {𝑝}) = (𝑢 “ {𝑝})))
5150adantr 472 . . . . . . . . . . . . . . . 16 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑤𝑈) → ((𝑤 “ {𝑝}) ∈ (𝑁𝑝) ↔ ∃𝑢𝑈 (𝑤 “ {𝑝}) = (𝑢 “ {𝑝})))
5246, 51mpbird 247 . . . . . . . . . . . . . . 15 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑤𝑈) → (𝑤 “ {𝑝}) ∈ (𝑁𝑝))
5335, 36, 40, 52syl21anc 1476 . . . . . . . . . . . . . 14 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ 𝑤𝑈) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎) → (𝑤 “ {𝑝}) ∈ (𝑁𝑝))
54 sseq1 3765 . . . . . . . . . . . . . . . . . 18 (𝑏 = (𝑤 “ {𝑝}) → (𝑏𝑎 ↔ (𝑤 “ {𝑝}) ⊆ 𝑎))
55543anbi2d 1551 . . . . . . . . . . . . . . . . 17 (𝑏 = (𝑤 “ {𝑝}) → (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑏𝑎𝑎𝑋) ↔ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎𝑎𝑋)))
56 eleq1 2825 . . . . . . . . . . . . . . . . 17 (𝑏 = (𝑤 “ {𝑝}) → (𝑏 ∈ (𝑁𝑝) ↔ (𝑤 “ {𝑝}) ∈ (𝑁𝑝)))
5755, 56anbi12d 749 . . . . . . . . . . . . . . . 16 (𝑏 = (𝑤 “ {𝑝}) → ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑏𝑎𝑎𝑋) ∧ 𝑏 ∈ (𝑁𝑝)) ↔ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎𝑎𝑋) ∧ (𝑤 “ {𝑝}) ∈ (𝑁𝑝))))
5857imbi1d 330 . . . . . . . . . . . . . . 15 (𝑏 = (𝑤 “ {𝑝}) → (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑏𝑎𝑎𝑋) ∧ 𝑏 ∈ (𝑁𝑝)) → 𝑎 ∈ (𝑁𝑝)) ↔ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎𝑎𝑋) ∧ (𝑤 “ {𝑝}) ∈ (𝑁𝑝)) → 𝑎 ∈ (𝑁𝑝))))
5913ustuqtop1 22244 . . . . . . . . . . . . . . 15 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑏𝑎𝑎𝑋) ∧ 𝑏 ∈ (𝑁𝑝)) → 𝑎 ∈ (𝑁𝑝))
6048, 58, 59vtocl 3397 . . . . . . . . . . . . . 14 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎𝑎𝑋) ∧ (𝑤 “ {𝑝}) ∈ (𝑁𝑝)) → 𝑎 ∈ (𝑁𝑝))
6137, 38, 39, 53, 60syl31anc 1480 . . . . . . . . . . . . 13 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ 𝑤𝑈) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎) → 𝑎 ∈ (𝑁𝑝))
6237, 21syl 17 . . . . . . . . . . . . 13 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ 𝑤𝑈) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎) → (𝑎 ∈ (𝑁𝑝) ↔ ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝})))
6361, 62mpbid 222 . . . . . . . . . . . 12 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ 𝑤𝑈) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎) → ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝}))
6463r19.29an 3213 . . . . . . . . . . 11 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ ∃𝑤𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎) → ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝}))
6534, 64impbida 913 . . . . . . . . . 10 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) → (∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝}) ↔ ∃𝑤𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎))
6622, 65bitrd 268 . . . . . . . . 9 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) → (𝑎 ∈ (𝑁𝑝) ↔ ∃𝑤𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎))
6766ralbidva 3121 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) → (∀𝑝𝑎 𝑎 ∈ (𝑁𝑝) ↔ ∀𝑝𝑎𝑤𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎))
6867rabbidva 3326 . . . . . . 7 (𝑈 ∈ (UnifOn‘𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)} = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎𝑤𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎})
693, 68eqtr4d 2795 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑋) → 𝐽 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)})
70 utopsnneip.1 . . . . . 6 𝐾 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)}
7169, 70syl6eqr 2810 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → 𝐽 = 𝐾)
7271fveq2d 6354 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (nei‘𝐽) = (nei‘𝐾))
7372fveq1d 6352 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → ((nei‘𝐽)‘{𝑃}) = ((nei‘𝐾)‘{𝑃}))
7473adantr 472 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → ((nei‘𝐽)‘{𝑃}) = ((nei‘𝐾)‘{𝑃}))
7513ustuqtop0 22243 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → 𝑁:𝑋⟶𝒫 𝒫 𝑋)
7613ustuqtop1 22244 . . . . 5 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → 𝑏 ∈ (𝑁𝑝))
7713ustuqtop2 22245 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (fi‘(𝑁𝑝)) ⊆ (𝑁𝑝))
7813ustuqtop3 22246 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → 𝑝𝑎)
7913ustuqtop4 22247 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → ∃𝑏 ∈ (𝑁𝑝)∀𝑞𝑏 𝑎 ∈ (𝑁𝑞))
8013ustuqtop5 22248 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → 𝑋 ∈ (𝑁𝑝))
8170, 75, 76, 77, 78, 79, 80neiptopnei 21136 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → 𝑁 = (𝑝𝑋 ↦ ((nei‘𝐾)‘{𝑝})))
8281adantr 472 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → 𝑁 = (𝑝𝑋 ↦ ((nei‘𝐾)‘{𝑝})))
83 simpr 479 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) ∧ 𝑝 = 𝑃) → 𝑝 = 𝑃)
8483sneqd 4331 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) ∧ 𝑝 = 𝑃) → {𝑝} = {𝑃})
8584fveq2d 6354 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) ∧ 𝑝 = 𝑃) → ((nei‘𝐾)‘{𝑝}) = ((nei‘𝐾)‘{𝑃}))
86 simpr 479 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → 𝑃𝑋)
87 fvexd 6362 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → ((nei‘𝐾)‘{𝑃}) ∈ V)
8882, 85, 86, 87fvmptd 6448 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → (𝑁𝑃) = ((nei‘𝐾)‘{𝑃}))
89 mptexg 6646 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
90 rnexg 7261 . . . . 5 ((𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V → ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
9189, 90syl 17 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
9291adantr 472 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
9313a1i 11 . . . 4 ((𝑃𝑋 ∧ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) → 𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝}))))
94 nfv 1990 . . . . . . . 8 𝑣 𝑃𝑋
95 nfmpt1 4897 . . . . . . . . . 10 𝑣(𝑣𝑈 ↦ (𝑣 “ {𝑃}))
9695nfrn 5521 . . . . . . . . 9 𝑣ran (𝑣𝑈 ↦ (𝑣 “ {𝑃}))
9796nfel1 2915 . . . . . . . 8 𝑣ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V
9894, 97nfan 1975 . . . . . . 7 𝑣(𝑃𝑋 ∧ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
99 nfv 1990 . . . . . . 7 𝑣 𝑝 = 𝑃
10098, 99nfan 1975 . . . . . 6 𝑣((𝑃𝑋 ∧ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) ∧ 𝑝 = 𝑃)
101 simpr2 1236 . . . . . . . . 9 ((𝑃𝑋 ∧ (ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V ∧ 𝑝 = 𝑃𝑣𝑈)) → 𝑝 = 𝑃)
102101sneqd 4331 . . . . . . . 8 ((𝑃𝑋 ∧ (ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V ∧ 𝑝 = 𝑃𝑣𝑈)) → {𝑝} = {𝑃})
103102imaeq2d 5622 . . . . . . 7 ((𝑃𝑋 ∧ (ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V ∧ 𝑝 = 𝑃𝑣𝑈)) → (𝑣 “ {𝑝}) = (𝑣 “ {𝑃}))
1041033anassrs 1454 . . . . . 6 ((((𝑃𝑋 ∧ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) ∧ 𝑝 = 𝑃) ∧ 𝑣𝑈) → (𝑣 “ {𝑝}) = (𝑣 “ {𝑃}))
105100, 104mpteq2da 4893 . . . . 5 (((𝑃𝑋 ∧ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) ∧ 𝑝 = 𝑃) → (𝑣𝑈 ↦ (𝑣 “ {𝑝})) = (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
106105rneqd 5506 . . . 4 (((𝑃𝑋 ∧ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) ∧ 𝑝 = 𝑃) → ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
107 simpl 474 . . . 4 ((𝑃𝑋 ∧ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) → 𝑃𝑋)
108 simpr 479 . . . 4 ((𝑃𝑋 ∧ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) → ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
10993, 106, 107, 108fvmptd 6448 . . 3 ((𝑃𝑋 ∧ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) → (𝑁𝑃) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
11086, 92, 109syl2anc 696 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → (𝑁𝑃) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
11174, 88, 1103eqtr2d 2798 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → ((nei‘𝐽)‘{𝑃}) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1072   = wceq 1630   ∈ wcel 2137  ∀wral 3048  ∃wrex 3049  {crab 3052  Vcvv 3338   ⊆ wss 3713  𝒫 cpw 4300  {csn 4319   ↦ cmpt 4879  ran crn 5265   “ cima 5267  ‘cfv 6047  neicnei 21101  UnifOncust 22202  unifTopcutop 22233 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-rep 4921  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-reu 3055  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-pss 3729  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-tp 4324  df-op 4326  df-uni 4587  df-int 4626  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-tr 4903  df-id 5172  df-eprel 5177  df-po 5185  df-so 5186  df-fr 5223  df-we 5225  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-pred 5839  df-ord 5885  df-on 5886  df-lim 5887  df-suc 5888  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-ov 6814  df-oprab 6815  df-mpt2 6816  df-om 7229  df-wrecs 7574  df-recs 7635  df-rdg 7673  df-1o 7727  df-oadd 7731  df-er 7909  df-en 8120  df-fin 8123  df-fi 8480  df-top 20899  df-nei 21102  df-ust 22203  df-utop 22234 This theorem is referenced by:  utopsnneip  22251
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