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Theorem utopsnneiplem 22938
 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 22923 . . . . . . . 8 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎𝑤𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎})
31, 2syl5eq 2806 . . . . . . 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 489 . . . . . . . . . . . . 13 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) → 𝑎 ∈ 𝒫 𝑋)
65elpwid 4503 . . . . . . . . . . . 12 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) → 𝑎𝑋)
76sselda 3893 . . . . . . . . . . 11 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) → 𝑝𝑋)
8 simpr 489 . . . . . . . . . . . . . 14 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → 𝑝𝑋)
9 mptexg 6973 . . . . . . . . . . . . . . . 16 (𝑈 ∈ (UnifOn‘𝑋) → (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ V)
10 rnexg 7612 . . . . . . . . . . . . . . . 16 ((𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ V → ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ V)
119, 10syl 17 . . . . . . . . . . . . . . 15 (𝑈 ∈ (UnifOn‘𝑋) → ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ V)
1211adantr 485 . . . . . . . . . . . . . 14 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ V)
13 utopsnneip.2 . . . . . . . . . . . . . . 15 𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
1413fvmpt2 6768 . . . . . . . . . . . . . 14 ((𝑝𝑋 ∧ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ V) → (𝑁𝑝) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
158, 12, 14syl2anc 588 . . . . . . . . . . . . 13 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (𝑁𝑝) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
1615eleq2d 2838 . . . . . . . . . . . 12 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (𝑎 ∈ (𝑁𝑝) ↔ 𝑎 ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝}))))
17 eqid 2759 . . . . . . . . . . . . . 14 (𝑣𝑈 ↦ (𝑣 “ {𝑝})) = (𝑣𝑈 ↦ (𝑣 “ {𝑝}))
1817elrnmpt 5795 . . . . . . . . . . . . 13 (𝑎 ∈ V → (𝑎 ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ↔ ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝})))
1918elv 3416 . . . . . . . . . . . 12 (𝑎 ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ↔ ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝}))
2016, 19bitrdi 290 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (𝑎 ∈ (𝑁𝑝) ↔ ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝})))
214, 7, 20syl2anc 588 . . . . . . . . . 10 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) → (𝑎 ∈ (𝑁𝑝) ↔ ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝})))
22 nfv 1916 . . . . . . . . . . . . 13 𝑣((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎)
23 nfre1 3231 . . . . . . . . . . . . 13 𝑣𝑣𝑈 𝑎 = (𝑣 “ {𝑝})
2422, 23nfan 1901 . . . . . . . . . . . 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 3950 . . . . . . . . . . . . . 14 (𝑎 = (𝑣 “ {𝑝}) → (𝑣 “ {𝑝}) ⊆ 𝑎)
2726adantl 486 . . . . . . . . . . . . 13 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝})) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑝})) → (𝑣 “ {𝑝}) ⊆ 𝑎)
28 imaeq1 5894 . . . . . . . . . . . . . . 15 (𝑤 = 𝑣 → (𝑤 “ {𝑝}) = (𝑣 “ {𝑝}))
2928sseq1d 3924 . . . . . . . . . . . . . 14 (𝑤 = 𝑣 → ((𝑤 “ {𝑝}) ⊆ 𝑎 ↔ (𝑣 “ {𝑝}) ⊆ 𝑎))
3029rspcev 3542 . . . . . . . . . . . . 13 ((𝑣𝑈 ∧ (𝑣 “ {𝑝}) ⊆ 𝑎) → ∃𝑤𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎)
3125, 27, 30syl2anc 588 . . . . . . . . . . . 12 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝})) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑝})) → ∃𝑤𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎)
32 simpr 489 . . . . . . . . . . . 12 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝})) → ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝}))
3324, 31, 32r19.29af 3255 . . . . . . . . . . 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 516 . . . . . . . . . . . . . 14 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ 𝑤𝑈) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎) → (𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋))
37 simpr 489 . . . . . . . . . . . . . 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 2759 . . . . . . . . . . . . . . . . . 18 (𝑤 “ {𝑝}) = (𝑤 “ {𝑝})
41 imaeq1 5894 . . . . . . . . . . . . . . . . . . 19 (𝑢 = 𝑤 → (𝑢 “ {𝑝}) = (𝑤 “ {𝑝}))
4241rspceeqv 3557 . . . . . . . . . . . . . . . . . 18 ((𝑤𝑈 ∧ (𝑤 “ {𝑝}) = (𝑤 “ {𝑝})) → ∃𝑢𝑈 (𝑤 “ {𝑝}) = (𝑢 “ {𝑝}))
4340, 42mpan2 691 . . . . . . . . . . . . . . . . 17 (𝑤𝑈 → ∃𝑢𝑈 (𝑤 “ {𝑝}) = (𝑢 “ {𝑝}))
4443adantl 486 . . . . . . . . . . . . . . . 16 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑤𝑈) → ∃𝑢𝑈 (𝑤 “ {𝑝}) = (𝑢 “ {𝑝}))
45 vex 3414 . . . . . . . . . . . . . . . . . . 19 𝑤 ∈ V
4645imaex 7624 . . . . . . . . . . . . . . . . . 18 (𝑤 “ {𝑝}) ∈ V
4713ustuqtoplem 22930 . . . . . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . . 16 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑤𝑈) → ((𝑤 “ {𝑝}) ∈ (𝑁𝑝) ↔ ∃𝑢𝑈 (𝑤 “ {𝑝}) = (𝑢 “ {𝑝})))
5044, 49mpbird 260 . . . . . . . . . . . . . . 15 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑤𝑈) → (𝑤 “ {𝑝}) ∈ (𝑁𝑝))
5134, 35, 39, 50syl21anc 837 . . . . . . . . . . . . . 14 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ 𝑤𝑈) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎) → (𝑤 “ {𝑝}) ∈ (𝑁𝑝))
52 sseq1 3918 . . . . . . . . . . . . . . . . . 18 (𝑏 = (𝑤 “ {𝑝}) → (𝑏𝑎 ↔ (𝑤 “ {𝑝}) ⊆ 𝑎))
53523anbi2d 1439 . . . . . . . . . . . . . . . . 17 (𝑏 = (𝑤 “ {𝑝}) → (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑏𝑎𝑎𝑋) ↔ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎𝑎𝑋)))
54 eleq1 2840 . . . . . . . . . . . . . . . . 17 (𝑏 = (𝑤 “ {𝑝}) → (𝑏 ∈ (𝑁𝑝) ↔ (𝑤 “ {𝑝}) ∈ (𝑁𝑝)))
5553, 54anbi12d 634 . . . . . . . . . . . . . . . 16 (𝑏 = (𝑤 “ {𝑝}) → ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑏𝑎𝑎𝑋) ∧ 𝑏 ∈ (𝑁𝑝)) ↔ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎𝑎𝑋) ∧ (𝑤 “ {𝑝}) ∈ (𝑁𝑝))))
5655imbi1d 346 . . . . . . . . . . . . . . 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 22932 . . . . . . . . . . . . . . 15 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑏𝑎𝑎𝑋) ∧ 𝑏 ∈ (𝑁𝑝)) → 𝑎 ∈ (𝑁𝑝))
5846, 56, 57vtocl 3478 . . . . . . . . . . . . . 14 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎𝑎𝑋) ∧ (𝑤 “ {𝑝}) ∈ (𝑁𝑝)) → 𝑎 ∈ (𝑁𝑝))
5936, 37, 38, 51, 58syl31anc 1371 . . . . . . . . . . . . 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 235 . . . . . . . . . . . 12 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ 𝑤𝑈) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎) → ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝}))
6261r19.29an 3213 . . . . . . . . . . 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 282 . . . . . . . . 9 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) → (𝑎 ∈ (𝑁𝑝) ↔ ∃𝑤𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎))
6564ralbidva 3126 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) → (∀𝑝𝑎 𝑎 ∈ (𝑁𝑝) ↔ ∀𝑝𝑎𝑤𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎))
6665rabbidva 3391 . . . . . . 7 (𝑈 ∈ (UnifOn‘𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)} = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎𝑤𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎})
673, 66eqtr4d 2797 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑋) → 𝐽 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)})
68 utopsnneip.1 . . . . . 6 𝐾 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)}
6967, 68eqtr4di 2812 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → 𝐽 = 𝐾)
7069fveq2d 6660 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (nei‘𝐽) = (nei‘𝐾))
7170fveq1d 6658 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → ((nei‘𝐽)‘{𝑃}) = ((nei‘𝐾)‘{𝑃}))
7271adantr 485 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → ((nei‘𝐽)‘{𝑃}) = ((nei‘𝐾)‘{𝑃}))
7313ustuqtop0 22931 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → 𝑁:𝑋⟶𝒫 𝒫 𝑋)
7413ustuqtop1 22932 . . . . 5 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → 𝑏 ∈ (𝑁𝑝))
7513ustuqtop2 22933 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (fi‘(𝑁𝑝)) ⊆ (𝑁𝑝))
7613ustuqtop3 22934 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → 𝑝𝑎)
7713ustuqtop4 22935 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → ∃𝑏 ∈ (𝑁𝑝)∀𝑞𝑏 𝑎 ∈ (𝑁𝑞))
7813ustuqtop5 22936 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → 𝑋 ∈ (𝑁𝑝))
7968, 73, 74, 75, 76, 77, 78neiptopnei 21822 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → 𝑁 = (𝑝𝑋 ↦ ((nei‘𝐾)‘{𝑝})))
8079adantr 485 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → 𝑁 = (𝑝𝑋 ↦ ((nei‘𝐾)‘{𝑝})))
81 simpr 489 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) ∧ 𝑝 = 𝑃) → 𝑝 = 𝑃)
8281sneqd 4532 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) ∧ 𝑝 = 𝑃) → {𝑝} = {𝑃})
8382fveq2d 6660 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) ∧ 𝑝 = 𝑃) → ((nei‘𝐾)‘{𝑝}) = ((nei‘𝐾)‘{𝑃}))
84 simpr 489 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → 𝑃𝑋)
85 fvexd 6671 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → ((nei‘𝐾)‘{𝑃}) ∈ V)
8680, 83, 84, 85fvmptd 6764 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → (𝑁𝑃) = ((nei‘𝐾)‘{𝑃}))
87 mptexg 6973 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
88 rnexg 7612 . . . . 5 ((𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V → ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
8987, 88syl 17 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
9089adantr 485 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
91 nfv 1916 . . . . . . . 8 𝑣 𝑃𝑋
92 nfmpt1 5128 . . . . . . . . . 10 𝑣(𝑣𝑈 ↦ (𝑣 “ {𝑃}))
9392nfrn 5791 . . . . . . . . 9 𝑣ran (𝑣𝑈 ↦ (𝑣 “ {𝑃}))
9493nfel1 2936 . . . . . . . 8 𝑣ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V
9591, 94nfan 1901 . . . . . . 7 𝑣(𝑃𝑋 ∧ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
96 nfv 1916 . . . . . . 7 𝑣 𝑝 = 𝑃
9795, 96nfan 1901 . . . . . 6 𝑣((𝑃𝑋 ∧ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) ∧ 𝑝 = 𝑃)
98 simpr2 1193 . . . . . . . . 9 ((𝑃𝑋 ∧ (ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V ∧ 𝑝 = 𝑃𝑣𝑈)) → 𝑝 = 𝑃)
9998sneqd 4532 . . . . . . . 8 ((𝑃𝑋 ∧ (ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V ∧ 𝑝 = 𝑃𝑣𝑈)) → {𝑝} = {𝑃})
10099imaeq2d 5899 . . . . . . 7 ((𝑃𝑋 ∧ (ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V ∧ 𝑝 = 𝑃𝑣𝑈)) → (𝑣 “ {𝑝}) = (𝑣 “ {𝑃}))
1011003anassrs 1358 . . . . . 6 ((((𝑃𝑋 ∧ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) ∧ 𝑝 = 𝑃) ∧ 𝑣𝑈) → (𝑣 “ {𝑝}) = (𝑣 “ {𝑃}))
10297, 101mpteq2da 5124 . . . . 5 (((𝑃𝑋 ∧ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) ∧ 𝑝 = 𝑃) → (𝑣𝑈 ↦ (𝑣 “ {𝑝})) = (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
103102rneqd 5777 . . . 4 (((𝑃𝑋 ∧ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) ∧ 𝑝 = 𝑃) → ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
104 simpl 487 . . . 4 ((𝑃𝑋 ∧ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) → 𝑃𝑋)
105 simpr 489 . . . 4 ((𝑃𝑋 ∧ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) → ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
10613, 103, 104, 105fvmptd2 6765 . . 3 ((𝑃𝑋 ∧ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) → (𝑁𝑃) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
10784, 90, 106syl2anc 588 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → (𝑁𝑃) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
10872, 86, 1073eqtr2d 2800 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → ((nei‘𝐽)‘{𝑃}) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1085   = wceq 1539   ∈ wcel 2112  ∀wral 3071  ∃wrex 3072  {crab 3075  Vcvv 3410   ⊆ wss 3859  𝒫 cpw 4492  {csn 4520   ↦ cmpt 5110  ran crn 5523   “ cima 5525  ‘cfv 6333  neicnei 21787  UnifOncust 22890  unifTopcutop 22921 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7457 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-pss 3878  df-nul 4227  df-if 4419  df-pw 4494  df-sn 4521  df-pr 4523  df-tp 4525  df-op 4527  df-uni 4797  df-int 4837  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5428  df-eprel 5433  df-po 5441  df-so 5442  df-fr 5481  df-we 5483  df-xp 5528  df-rel 5529  df-cnv 5530  df-co 5531  df-dm 5532  df-rn 5533  df-res 5534  df-ima 5535  df-pred 6124  df-ord 6170  df-on 6171  df-lim 6172  df-suc 6173  df-iota 6292  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7578  df-wrecs 7955  df-recs 8016  df-rdg 8054  df-1o 8110  df-oadd 8114  df-er 8297  df-en 8526  df-fin 8529  df-fi 8898  df-top 21584  df-nei 21788  df-ust 22891  df-utop 22922 This theorem is referenced by:  utopsnneip  22939
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