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Theorem utopsnneiplem 22098
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 22083 . . . . . . . 8 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎𝑤𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎})
31, 2syl5eq 2697 . . . . . . 7 (𝑈 ∈ (UnifOn‘𝑋) → 𝐽 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎𝑤𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎})
4 simpll 805 . . . . . . . . . . 11 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) → 𝑈 ∈ (UnifOn‘𝑋))
5 simpr 476 . . . . . . . . . . . . 13 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) → 𝑎 ∈ 𝒫 𝑋)
65elpwid 4203 . . . . . . . . . . . 12 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) → 𝑎𝑋)
76sselda 3636 . . . . . . . . . . 11 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) → 𝑝𝑋)
8 simpr 476 . . . . . . . . . . . . . 14 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → 𝑝𝑋)
9 mptexg 6525 . . . . . . . . . . . . . . . 16 (𝑈 ∈ (UnifOn‘𝑋) → (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ V)
10 rnexg 7140 . . . . . . . . . . . . . . . 16 ((𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ V → ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ V)
119, 10syl 17 . . . . . . . . . . . . . . 15 (𝑈 ∈ (UnifOn‘𝑋) → ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ V)
1211adantr 480 . . . . . . . . . . . . . 14 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ V)
13 utopsnneip.2 . . . . . . . . . . . . . . 15 𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
1413fvmpt2 6330 . . . . . . . . . . . . . 14 ((𝑝𝑋 ∧ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ V) → (𝑁𝑝) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
158, 12, 14syl2anc 694 . . . . . . . . . . . . 13 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (𝑁𝑝) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
1615eleq2d 2716 . . . . . . . . . . . 12 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (𝑎 ∈ (𝑁𝑝) ↔ 𝑎 ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝}))))
17 vex 3234 . . . . . . . . . . . . 13 𝑎 ∈ V
18 eqid 2651 . . . . . . . . . . . . . 14 (𝑣𝑈 ↦ (𝑣 “ {𝑝})) = (𝑣𝑈 ↦ (𝑣 “ {𝑝}))
1918elrnmpt 5404 . . . . . . . . . . . . 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 694 . . . . . . . . . 10 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) → (𝑎 ∈ (𝑁𝑝) ↔ ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝})))
23 nfv 1883 . . . . . . . . . . . . 13 𝑣((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎)
24 nfre1 3034 . . . . . . . . . . . . 13 𝑣𝑣𝑈 𝑎 = (𝑣 “ {𝑝})
2523, 24nfan 1868 . . . . . . . . . . . 12 𝑣(((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝}))
26 simplr 807 . . . . . . . . . . . . 13 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝})) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑝})) → 𝑣𝑈)
27 eqimss2 3691 . . . . . . . . . . . . . 14 (𝑎 = (𝑣 “ {𝑝}) → (𝑣 “ {𝑝}) ⊆ 𝑎)
2827adantl 481 . . . . . . . . . . . . 13 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝})) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑝})) → (𝑣 “ {𝑝}) ⊆ 𝑎)
29 imaeq1 5496 . . . . . . . . . . . . . . 15 (𝑤 = 𝑣 → (𝑤 “ {𝑝}) = (𝑣 “ {𝑝}))
3029sseq1d 3665 . . . . . . . . . . . . . 14 (𝑤 = 𝑣 → ((𝑤 “ {𝑝}) ⊆ 𝑎 ↔ (𝑣 “ {𝑝}) ⊆ 𝑎))
3130rspcev 3340 . . . . . . . . . . . . 13 ((𝑣𝑈 ∧ (𝑣 “ {𝑝}) ⊆ 𝑎) → ∃𝑤𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎)
3226, 28, 31syl2anc 694 . . . . . . . . . . . 12 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝})) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑝})) → ∃𝑤𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎)
33 simpr 476 . . . . . . . . . . . 12 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝})) → ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝}))
3425, 32, 33r19.29af 3105 . . . . . . . . . . 11 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝})) → ∃𝑤𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎)
354ad2antrr 762 . . . . . . . . . . . . . . 15 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ 𝑤𝑈) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎) → 𝑈 ∈ (UnifOn‘𝑋))
367ad2antrr 762 . . . . . . . . . . . . . . 15 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ 𝑤𝑈) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎) → 𝑝𝑋)
3735, 36jca 553 . . . . . . . . . . . . . 14 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ 𝑤𝑈) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎) → (𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋))
38 simpr 476 . . . . . . . . . . . . . 14 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ 𝑤𝑈) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎) → (𝑤 “ {𝑝}) ⊆ 𝑎)
396ad3antrrr 766 . . . . . . . . . . . . . 14 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ 𝑤𝑈) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎) → 𝑎𝑋)
40 simplr 807 . . . . . . . . . . . . . . 15 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ 𝑤𝑈) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎) → 𝑤𝑈)
41 eqid 2651 . . . . . . . . . . . . . . . . . 18 (𝑤 “ {𝑝}) = (𝑤 “ {𝑝})
42 imaeq1 5496 . . . . . . . . . . . . . . . . . . . 20 (𝑢 = 𝑤 → (𝑢 “ {𝑝}) = (𝑤 “ {𝑝}))
4342eqeq2d 2661 . . . . . . . . . . . . . . . . . . 19 (𝑢 = 𝑤 → ((𝑤 “ {𝑝}) = (𝑢 “ {𝑝}) ↔ (𝑤 “ {𝑝}) = (𝑤 “ {𝑝})))
4443rspcev 3340 . . . . . . . . . . . . . . . . . 18 ((𝑤𝑈 ∧ (𝑤 “ {𝑝}) = (𝑤 “ {𝑝})) → ∃𝑢𝑈 (𝑤 “ {𝑝}) = (𝑢 “ {𝑝}))
4541, 44mpan2 707 . . . . . . . . . . . . . . . . 17 (𝑤𝑈 → ∃𝑢𝑈 (𝑤 “ {𝑝}) = (𝑢 “ {𝑝}))
4645adantl 481 . . . . . . . . . . . . . . . 16 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑤𝑈) → ∃𝑢𝑈 (𝑤 “ {𝑝}) = (𝑢 “ {𝑝}))
47 vex 3234 . . . . . . . . . . . . . . . . . . 19 𝑤 ∈ V
4847imaex 7146 . . . . . . . . . . . . . . . . . 18 (𝑤 “ {𝑝}) ∈ V
4913ustuqtoplem 22090 . . . . . . . . . . . . . . . . . 18 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ (𝑤 “ {𝑝}) ∈ V) → ((𝑤 “ {𝑝}) ∈ (𝑁𝑝) ↔ ∃𝑢𝑈 (𝑤 “ {𝑝}) = (𝑢 “ {𝑝})))
5048, 49mpan2 707 . . . . . . . . . . . . . . . . 17 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → ((𝑤 “ {𝑝}) ∈ (𝑁𝑝) ↔ ∃𝑢𝑈 (𝑤 “ {𝑝}) = (𝑢 “ {𝑝})))
5150adantr 480 . . . . . . . . . . . . . . . 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 1365 . . . . . . . . . . . . . 14 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ 𝑤𝑈) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎) → (𝑤 “ {𝑝}) ∈ (𝑁𝑝))
54 sseq1 3659 . . . . . . . . . . . . . . . . . 18 (𝑏 = (𝑤 “ {𝑝}) → (𝑏𝑎 ↔ (𝑤 “ {𝑝}) ⊆ 𝑎))
55543anbi2d 1444 . . . . . . . . . . . . . . . . 17 (𝑏 = (𝑤 “ {𝑝}) → (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑏𝑎𝑎𝑋) ↔ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎𝑎𝑋)))
56 eleq1 2718 . . . . . . . . . . . . . . . . 17 (𝑏 = (𝑤 “ {𝑝}) → (𝑏 ∈ (𝑁𝑝) ↔ (𝑤 “ {𝑝}) ∈ (𝑁𝑝)))
5755, 56anbi12d 747 . . . . . . . . . . . . . . . 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 22092 . . . . . . . . . . . . . . 15 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑏𝑎𝑎𝑋) ∧ 𝑏 ∈ (𝑁𝑝)) → 𝑎 ∈ (𝑁𝑝))
6048, 58, 59vtocl 3290 . . . . . . . . . . . . . 14 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎𝑎𝑋) ∧ (𝑤 “ {𝑝}) ∈ (𝑁𝑝)) → 𝑎 ∈ (𝑁𝑝))
6137, 38, 39, 53, 60syl31anc 1369 . . . . . . . . . . . . 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 3106 . . . . . . . . . . 11 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ ∃𝑤𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎) → ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝}))
6534, 64impbida 895 . . . . . . . . . 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 3014 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) → (∀𝑝𝑎 𝑎 ∈ (𝑁𝑝) ↔ ∀𝑝𝑎𝑤𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎))
6867rabbidva 3219 . . . . . . 7 (𝑈 ∈ (UnifOn‘𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)} = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎𝑤𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎})
693, 68eqtr4d 2688 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑋) → 𝐽 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)})
70 utopsnneip.1 . . . . . 6 𝐾 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)}
7169, 70syl6eqr 2703 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → 𝐽 = 𝐾)
7271fveq2d 6233 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (nei‘𝐽) = (nei‘𝐾))
7372fveq1d 6231 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → ((nei‘𝐽)‘{𝑃}) = ((nei‘𝐾)‘{𝑃}))
7473adantr 480 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → ((nei‘𝐽)‘{𝑃}) = ((nei‘𝐾)‘{𝑃}))
7513ustuqtop0 22091 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → 𝑁:𝑋⟶𝒫 𝒫 𝑋)
7613ustuqtop1 22092 . . . . 5 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → 𝑏 ∈ (𝑁𝑝))
7713ustuqtop2 22093 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (fi‘(𝑁𝑝)) ⊆ (𝑁𝑝))
7813ustuqtop3 22094 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → 𝑝𝑎)
7913ustuqtop4 22095 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → ∃𝑏 ∈ (𝑁𝑝)∀𝑞𝑏 𝑎 ∈ (𝑁𝑞))
8013ustuqtop5 22096 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → 𝑋 ∈ (𝑁𝑝))
8170, 75, 76, 77, 78, 79, 80neiptopnei 20984 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → 𝑁 = (𝑝𝑋 ↦ ((nei‘𝐾)‘{𝑝})))
8281adantr 480 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → 𝑁 = (𝑝𝑋 ↦ ((nei‘𝐾)‘{𝑝})))
83 simpr 476 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) ∧ 𝑝 = 𝑃) → 𝑝 = 𝑃)
8483sneqd 4222 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) ∧ 𝑝 = 𝑃) → {𝑝} = {𝑃})
8584fveq2d 6233 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) ∧ 𝑝 = 𝑃) → ((nei‘𝐾)‘{𝑝}) = ((nei‘𝐾)‘{𝑃}))
86 simpr 476 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → 𝑃𝑋)
87 fvexd 6241 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → ((nei‘𝐾)‘{𝑃}) ∈ V)
8882, 85, 86, 87fvmptd 6327 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → (𝑁𝑃) = ((nei‘𝐾)‘{𝑃}))
89 mptexg 6525 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
90 rnexg 7140 . . . . 5 ((𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V → ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
9189, 90syl 17 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
9291adantr 480 . . 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 1883 . . . . . . . 8 𝑣 𝑃𝑋
95 nfmpt1 4780 . . . . . . . . . 10 𝑣(𝑣𝑈 ↦ (𝑣 “ {𝑃}))
9695nfrn 5400 . . . . . . . . 9 𝑣ran (𝑣𝑈 ↦ (𝑣 “ {𝑃}))
9796nfel1 2808 . . . . . . . 8 𝑣ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V
9894, 97nfan 1868 . . . . . . 7 𝑣(𝑃𝑋 ∧ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
99 nfv 1883 . . . . . . 7 𝑣 𝑝 = 𝑃
10098, 99nfan 1868 . . . . . 6 𝑣((𝑃𝑋 ∧ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) ∧ 𝑝 = 𝑃)
101 simpr2 1088 . . . . . . . . 9 ((𝑃𝑋 ∧ (ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V ∧ 𝑝 = 𝑃𝑣𝑈)) → 𝑝 = 𝑃)
102101sneqd 4222 . . . . . . . 8 ((𝑃𝑋 ∧ (ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V ∧ 𝑝 = 𝑃𝑣𝑈)) → {𝑝} = {𝑃})
103102imaeq2d 5501 . . . . . . 7 ((𝑃𝑋 ∧ (ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V ∧ 𝑝 = 𝑃𝑣𝑈)) → (𝑣 “ {𝑝}) = (𝑣 “ {𝑃}))
1041033anassrs 1313 . . . . . 6 ((((𝑃𝑋 ∧ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) ∧ 𝑝 = 𝑃) ∧ 𝑣𝑈) → (𝑣 “ {𝑝}) = (𝑣 “ {𝑃}))
105100, 104mpteq2da 4776 . . . . 5 (((𝑃𝑋 ∧ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) ∧ 𝑝 = 𝑃) → (𝑣𝑈 ↦ (𝑣 “ {𝑝})) = (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
106105rneqd 5385 . . . 4 (((𝑃𝑋 ∧ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) ∧ 𝑝 = 𝑃) → ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
107 simpl 472 . . . 4 ((𝑃𝑋 ∧ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) → 𝑃𝑋)
108 simpr 476 . . . 4 ((𝑃𝑋 ∧ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) → ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
10993, 106, 107, 108fvmptd 6327 . . 3 ((𝑃𝑋 ∧ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) → (𝑁𝑃) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
11086, 92, 109syl2anc 694 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → (𝑁𝑃) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
11174, 88, 1103eqtr2d 2691 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 1054   = wceq 1523  wcel 2030  wral 2941  wrex 2942  {crab 2945  Vcvv 3231  wss 3607  𝒫 cpw 4191  {csn 4210  cmpt 4762  ran crn 5144  cima 5146  cfv 5926  neicnei 20949  UnifOncust 22050  unifTopcutop 22081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-fin 8001  df-fi 8358  df-top 20747  df-nei 20950  df-ust 22051  df-utop 22082
This theorem is referenced by:  utopsnneip  22099
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