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Theorem utopsnneiplem 24272
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 24257 . . . . . . . 8 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎𝑤𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎})
31, 2eqtrid 2787 . . . . . . 7 (𝑈 ∈ (UnifOn‘𝑋) → 𝐽 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎𝑤𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎})
4 simpll 767 . . . . . . . . . . 11 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) → 𝑈 ∈ (UnifOn‘𝑋))
5 simpr 484 . . . . . . . . . . . . 13 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) → 𝑎 ∈ 𝒫 𝑋)
65elpwid 4614 . . . . . . . . . . . 12 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) → 𝑎𝑋)
76sselda 3995 . . . . . . . . . . 11 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) → 𝑝𝑋)
8 simpr 484 . . . . . . . . . . . . . 14 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → 𝑝𝑋)
9 mptexg 7241 . . . . . . . . . . . . . . . 16 (𝑈 ∈ (UnifOn‘𝑋) → (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ V)
10 rnexg 7925 . . . . . . . . . . . . . . . 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 7027 . . . . . . . . . . . . . 14 ((𝑝𝑋 ∧ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ V) → (𝑁𝑝) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
158, 12, 14syl2anc 584 . . . . . . . . . . . . 13 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (𝑁𝑝) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
1615eleq2d 2825 . . . . . . . . . . . 12 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (𝑎 ∈ (𝑁𝑝) ↔ 𝑎 ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝}))))
17 eqid 2735 . . . . . . . . . . . . . 14 (𝑣𝑈 ↦ (𝑣 “ {𝑝})) = (𝑣𝑈 ↦ (𝑣 “ {𝑝}))
1817elrnmpt 5972 . . . . . . . . . . . . 13 (𝑎 ∈ V → (𝑎 ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ↔ ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝})))
1918elv 3483 . . . . . . . . . . . 12 (𝑎 ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ↔ ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝}))
2016, 19bitrdi 287 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (𝑎 ∈ (𝑁𝑝) ↔ ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝})))
214, 7, 20syl2anc 584 . . . . . . . . . 10 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) → (𝑎 ∈ (𝑁𝑝) ↔ ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝})))
22 nfv 1912 . . . . . . . . . . . . 13 𝑣((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎)
23 nfre1 3283 . . . . . . . . . . . . 13 𝑣𝑣𝑈 𝑎 = (𝑣 “ {𝑝})
2422, 23nfan 1897 . . . . . . . . . . . 12 𝑣(((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝}))
25 simplr 769 . . . . . . . . . . . . 13 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝})) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑝})) → 𝑣𝑈)
26 eqimss2 4055 . . . . . . . . . . . . . 14 (𝑎 = (𝑣 “ {𝑝}) → (𝑣 “ {𝑝}) ⊆ 𝑎)
2726adantl 481 . . . . . . . . . . . . 13 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝})) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑝})) → (𝑣 “ {𝑝}) ⊆ 𝑎)
28 imaeq1 6075 . . . . . . . . . . . . . . 15 (𝑤 = 𝑣 → (𝑤 “ {𝑝}) = (𝑣 “ {𝑝}))
2928sseq1d 4027 . . . . . . . . . . . . . 14 (𝑤 = 𝑣 → ((𝑤 “ {𝑝}) ⊆ 𝑎 ↔ (𝑣 “ {𝑝}) ⊆ 𝑎))
3029rspcev 3622 . . . . . . . . . . . . 13 ((𝑣𝑈 ∧ (𝑣 “ {𝑝}) ⊆ 𝑎) → ∃𝑤𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎)
3125, 27, 30syl2anc 584 . . . . . . . . . . . 12 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝})) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑝})) → ∃𝑤𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎)
32 simpr 484 . . . . . . . . . . . 12 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝})) → ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝}))
3324, 31, 32r19.29af 3266 . . . . . . . . . . 11 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝})) → ∃𝑤𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎)
344ad2antrr 726 . . . . . . . . . . . . . . 15 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ 𝑤𝑈) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎) → 𝑈 ∈ (UnifOn‘𝑋))
357ad2antrr 726 . . . . . . . . . . . . . . 15 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ 𝑤𝑈) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎) → 𝑝𝑋)
3634, 35jca 511 . . . . . . . . . . . . . 14 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ 𝑤𝑈) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎) → (𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋))
37 simpr 484 . . . . . . . . . . . . . 14 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ 𝑤𝑈) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎) → (𝑤 “ {𝑝}) ⊆ 𝑎)
386ad3antrrr 730 . . . . . . . . . . . . . 14 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ 𝑤𝑈) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎) → 𝑎𝑋)
39 simplr 769 . . . . . . . . . . . . . . 15 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ 𝑤𝑈) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎) → 𝑤𝑈)
40 eqid 2735 . . . . . . . . . . . . . . . . . 18 (𝑤 “ {𝑝}) = (𝑤 “ {𝑝})
41 imaeq1 6075 . . . . . . . . . . . . . . . . . . 19 (𝑢 = 𝑤 → (𝑢 “ {𝑝}) = (𝑤 “ {𝑝}))
4241rspceeqv 3645 . . . . . . . . . . . . . . . . . 18 ((𝑤𝑈 ∧ (𝑤 “ {𝑝}) = (𝑤 “ {𝑝})) → ∃𝑢𝑈 (𝑤 “ {𝑝}) = (𝑢 “ {𝑝}))
4340, 42mpan2 691 . . . . . . . . . . . . . . . . 17 (𝑤𝑈 → ∃𝑢𝑈 (𝑤 “ {𝑝}) = (𝑢 “ {𝑝}))
4443adantl 481 . . . . . . . . . . . . . . . 16 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑤𝑈) → ∃𝑢𝑈 (𝑤 “ {𝑝}) = (𝑢 “ {𝑝}))
45 vex 3482 . . . . . . . . . . . . . . . . . . 19 𝑤 ∈ V
4645imaex 7937 . . . . . . . . . . . . . . . . . 18 (𝑤 “ {𝑝}) ∈ V
4713ustuqtoplem 24264 . . . . . . . . . . . . . . . . . 18 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ (𝑤 “ {𝑝}) ∈ V) → ((𝑤 “ {𝑝}) ∈ (𝑁𝑝) ↔ ∃𝑢𝑈 (𝑤 “ {𝑝}) = (𝑢 “ {𝑝})))
4846, 47mpan2 691 . . . . . . . . . . . . . . . . 17 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → ((𝑤 “ {𝑝}) ∈ (𝑁𝑝) ↔ ∃𝑢𝑈 (𝑤 “ {𝑝}) = (𝑢 “ {𝑝})))
4948adantr 480 . . . . . . . . . . . . . . . 16 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑤𝑈) → ((𝑤 “ {𝑝}) ∈ (𝑁𝑝) ↔ ∃𝑢𝑈 (𝑤 “ {𝑝}) = (𝑢 “ {𝑝})))
5044, 49mpbird 257 . . . . . . . . . . . . . . 15 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑤𝑈) → (𝑤 “ {𝑝}) ∈ (𝑁𝑝))
5134, 35, 39, 50syl21anc 838 . . . . . . . . . . . . . 14 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ 𝑤𝑈) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎) → (𝑤 “ {𝑝}) ∈ (𝑁𝑝))
52 sseq1 4021 . . . . . . . . . . . . . . . . . 18 (𝑏 = (𝑤 “ {𝑝}) → (𝑏𝑎 ↔ (𝑤 “ {𝑝}) ⊆ 𝑎))
53523anbi2d 1440 . . . . . . . . . . . . . . . . 17 (𝑏 = (𝑤 “ {𝑝}) → (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑏𝑎𝑎𝑋) ↔ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎𝑎𝑋)))
54 eleq1 2827 . . . . . . . . . . . . . . . . 17 (𝑏 = (𝑤 “ {𝑝}) → (𝑏 ∈ (𝑁𝑝) ↔ (𝑤 “ {𝑝}) ∈ (𝑁𝑝)))
5553, 54anbi12d 632 . . . . . . . . . . . . . . . 16 (𝑏 = (𝑤 “ {𝑝}) → ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑏𝑎𝑎𝑋) ∧ 𝑏 ∈ (𝑁𝑝)) ↔ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎𝑎𝑋) ∧ (𝑤 “ {𝑝}) ∈ (𝑁𝑝))))
5655imbi1d 341 . . . . . . . . . . . . . . 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))))
5713ustuqtop1 24266 . . . . . . . . . . . . . . 15 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑏𝑎𝑎𝑋) ∧ 𝑏 ∈ (𝑁𝑝)) → 𝑎 ∈ (𝑁𝑝))
5846, 56, 57vtocl 3558 . . . . . . . . . . . . . 14 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎𝑎𝑋) ∧ (𝑤 “ {𝑝}) ∈ (𝑁𝑝)) → 𝑎 ∈ (𝑁𝑝))
5936, 37, 38, 51, 58syl31anc 1372 . . . . . . . . . . . . 13 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ 𝑤𝑈) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎) → 𝑎 ∈ (𝑁𝑝))
6036, 20syl 17 . . . . . . . . . . . . 13 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ 𝑤𝑈) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎) → (𝑎 ∈ (𝑁𝑝) ↔ ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝})))
6159, 60mpbid 232 . . . . . . . . . . . 12 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ 𝑤𝑈) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎) → ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝}))
6261r19.29an 3156 . . . . . . . . . . 11 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ ∃𝑤𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎) → ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝}))
6333, 62impbida 801 . . . . . . . . . 10 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) → (∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝}) ↔ ∃𝑤𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎))
6421, 63bitrd 279 . . . . . . . . 9 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) → (𝑎 ∈ (𝑁𝑝) ↔ ∃𝑤𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎))
6564ralbidva 3174 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) → (∀𝑝𝑎 𝑎 ∈ (𝑁𝑝) ↔ ∀𝑝𝑎𝑤𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎))
6665rabbidva 3440 . . . . . . 7 (𝑈 ∈ (UnifOn‘𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)} = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎𝑤𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎})
673, 66eqtr4d 2778 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑋) → 𝐽 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)})
68 utopsnneip.1 . . . . . 6 𝐾 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)}
6967, 68eqtr4di 2793 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → 𝐽 = 𝐾)
7069fveq2d 6911 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (nei‘𝐽) = (nei‘𝐾))
7170fveq1d 6909 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → ((nei‘𝐽)‘{𝑃}) = ((nei‘𝐾)‘{𝑃}))
7271adantr 480 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → ((nei‘𝐽)‘{𝑃}) = ((nei‘𝐾)‘{𝑃}))
7313ustuqtop0 24265 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → 𝑁:𝑋⟶𝒫 𝒫 𝑋)
7413ustuqtop1 24266 . . . . 5 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → 𝑏 ∈ (𝑁𝑝))
7513ustuqtop2 24267 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (fi‘(𝑁𝑝)) ⊆ (𝑁𝑝))
7613ustuqtop3 24268 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → 𝑝𝑎)
7713ustuqtop4 24269 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → ∃𝑏 ∈ (𝑁𝑝)∀𝑞𝑏 𝑎 ∈ (𝑁𝑞))
7813ustuqtop5 24270 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → 𝑋 ∈ (𝑁𝑝))
7968, 73, 74, 75, 76, 77, 78neiptopnei 23156 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → 𝑁 = (𝑝𝑋 ↦ ((nei‘𝐾)‘{𝑝})))
8079adantr 480 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → 𝑁 = (𝑝𝑋 ↦ ((nei‘𝐾)‘{𝑝})))
81 simpr 484 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) ∧ 𝑝 = 𝑃) → 𝑝 = 𝑃)
8281sneqd 4643 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) ∧ 𝑝 = 𝑃) → {𝑝} = {𝑃})
8382fveq2d 6911 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) ∧ 𝑝 = 𝑃) → ((nei‘𝐾)‘{𝑝}) = ((nei‘𝐾)‘{𝑃}))
84 simpr 484 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → 𝑃𝑋)
85 fvexd 6922 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → ((nei‘𝐾)‘{𝑃}) ∈ V)
8680, 83, 84, 85fvmptd 7023 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → (𝑁𝑃) = ((nei‘𝐾)‘{𝑃}))
87 mptexg 7241 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
88 rnexg 7925 . . . . 5 ((𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V → ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
8987, 88syl 17 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
9089adantr 480 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
91 nfv 1912 . . . . . . . 8 𝑣 𝑃𝑋
92 nfmpt1 5256 . . . . . . . . . 10 𝑣(𝑣𝑈 ↦ (𝑣 “ {𝑃}))
9392nfrn 5966 . . . . . . . . 9 𝑣ran (𝑣𝑈 ↦ (𝑣 “ {𝑃}))
9493nfel1 2920 . . . . . . . 8 𝑣ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V
9591, 94nfan 1897 . . . . . . 7 𝑣(𝑃𝑋 ∧ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
96 nfv 1912 . . . . . . 7 𝑣 𝑝 = 𝑃
9795, 96nfan 1897 . . . . . 6 𝑣((𝑃𝑋 ∧ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) ∧ 𝑝 = 𝑃)
98 simpr2 1194 . . . . . . . . 9 ((𝑃𝑋 ∧ (ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V ∧ 𝑝 = 𝑃𝑣𝑈)) → 𝑝 = 𝑃)
9998sneqd 4643 . . . . . . . 8 ((𝑃𝑋 ∧ (ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V ∧ 𝑝 = 𝑃𝑣𝑈)) → {𝑝} = {𝑃})
10099imaeq2d 6080 . . . . . . 7 ((𝑃𝑋 ∧ (ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V ∧ 𝑝 = 𝑃𝑣𝑈)) → (𝑣 “ {𝑝}) = (𝑣 “ {𝑃}))
1011003anassrs 1359 . . . . . 6 ((((𝑃𝑋 ∧ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) ∧ 𝑝 = 𝑃) ∧ 𝑣𝑈) → (𝑣 “ {𝑝}) = (𝑣 “ {𝑃}))
10297, 101mpteq2da 5246 . . . . 5 (((𝑃𝑋 ∧ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) ∧ 𝑝 = 𝑃) → (𝑣𝑈 ↦ (𝑣 “ {𝑝})) = (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
103102rneqd 5952 . . . 4 (((𝑃𝑋 ∧ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) ∧ 𝑝 = 𝑃) → ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
104 simpl 482 . . . 4 ((𝑃𝑋 ∧ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) → 𝑃𝑋)
105 simpr 484 . . . 4 ((𝑃𝑋 ∧ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) → ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
10613, 103, 104, 105fvmptd2 7024 . . 3 ((𝑃𝑋 ∧ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) → (𝑁𝑃) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
10784, 90, 106syl2anc 584 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → (𝑁𝑃) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
10872, 86, 1073eqtr2d 2781 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → ((nei‘𝐽)‘{𝑃}) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  wrex 3068  {crab 3433  Vcvv 3478  wss 3963  𝒫 cpw 4605  {csn 4631  cmpt 5231  ran crn 5690  cima 5692  cfv 6563  neicnei 23121  UnifOncust 24224  unifTopcutop 24255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-om 7888  df-1o 8505  df-2o 8506  df-en 8985  df-fin 8988  df-fi 9449  df-top 22916  df-nei 23122  df-ust 24225  df-utop 24256
This theorem is referenced by:  utopsnneip  24273
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