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Theorem utopsnneiplem 21961
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 21946 . . . . . . . 8 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎𝑤𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎})
31, 2syl5eq 2667 . . . . . . 7 (𝑈 ∈ (UnifOn‘𝑋) → 𝐽 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎𝑤𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎})
4 simpll 789 . . . . . . . . . . 11 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) → 𝑈 ∈ (UnifOn‘𝑋))
5 simpr 477 . . . . . . . . . . . . 13 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) → 𝑎 ∈ 𝒫 𝑋)
65elpwid 4141 . . . . . . . . . . . 12 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) → 𝑎𝑋)
76sselda 3583 . . . . . . . . . . 11 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) → 𝑝𝑋)
8 simpr 477 . . . . . . . . . . . . . 14 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → 𝑝𝑋)
9 mptexg 6438 . . . . . . . . . . . . . . . 16 (𝑈 ∈ (UnifOn‘𝑋) → (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ V)
10 rnexg 7045 . . . . . . . . . . . . . . . 16 ((𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ V → ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ V)
119, 10syl 17 . . . . . . . . . . . . . . 15 (𝑈 ∈ (UnifOn‘𝑋) → ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ V)
1211adantr 481 . . . . . . . . . . . . . 14 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ V)
13 utopsnneip.2 . . . . . . . . . . . . . . 15 𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
1413fvmpt2 6248 . . . . . . . . . . . . . 14 ((𝑝𝑋 ∧ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ V) → (𝑁𝑝) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
158, 12, 14syl2anc 692 . . . . . . . . . . . . 13 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (𝑁𝑝) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
1615eleq2d 2684 . . . . . . . . . . . 12 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (𝑎 ∈ (𝑁𝑝) ↔ 𝑎 ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝}))))
17 vex 3189 . . . . . . . . . . . . 13 𝑎 ∈ V
18 eqid 2621 . . . . . . . . . . . . . 14 (𝑣𝑈 ↦ (𝑣 “ {𝑝})) = (𝑣𝑈 ↦ (𝑣 “ {𝑝}))
1918elrnmpt 5332 . . . . . . . . . . . . 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 692 . . . . . . . . . 10 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) → (𝑎 ∈ (𝑁𝑝) ↔ ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝})))
23 nfv 1840 . . . . . . . . . . . . 13 𝑣((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎)
24 nfre1 2999 . . . . . . . . . . . . 13 𝑣𝑣𝑈 𝑎 = (𝑣 “ {𝑝})
2523, 24nfan 1825 . . . . . . . . . . . 12 𝑣(((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝}))
26 simplr 791 . . . . . . . . . . . . 13 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝})) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑝})) → 𝑣𝑈)
27 eqimss2 3637 . . . . . . . . . . . . . 14 (𝑎 = (𝑣 “ {𝑝}) → (𝑣 “ {𝑝}) ⊆ 𝑎)
2827adantl 482 . . . . . . . . . . . . 13 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝})) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑝})) → (𝑣 “ {𝑝}) ⊆ 𝑎)
29 imaeq1 5420 . . . . . . . . . . . . . . 15 (𝑤 = 𝑣 → (𝑤 “ {𝑝}) = (𝑣 “ {𝑝}))
3029sseq1d 3611 . . . . . . . . . . . . . 14 (𝑤 = 𝑣 → ((𝑤 “ {𝑝}) ⊆ 𝑎 ↔ (𝑣 “ {𝑝}) ⊆ 𝑎))
3130rspcev 3295 . . . . . . . . . . . . 13 ((𝑣𝑈 ∧ (𝑣 “ {𝑝}) ⊆ 𝑎) → ∃𝑤𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎)
3226, 28, 31syl2anc 692 . . . . . . . . . . . 12 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝})) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑝})) → ∃𝑤𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎)
33 simpr 477 . . . . . . . . . . . 12 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝})) → ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝}))
3425, 32, 33r19.29af 3069 . . . . . . . . . . 11 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝})) → ∃𝑤𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎)
354ad2antrr 761 . . . . . . . . . . . . . . 15 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ 𝑤𝑈) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎) → 𝑈 ∈ (UnifOn‘𝑋))
367ad2antrr 761 . . . . . . . . . . . . . . 15 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ 𝑤𝑈) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎) → 𝑝𝑋)
3735, 36jca 554 . . . . . . . . . . . . . 14 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ 𝑤𝑈) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎) → (𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋))
38 simpr 477 . . . . . . . . . . . . . 14 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ 𝑤𝑈) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎) → (𝑤 “ {𝑝}) ⊆ 𝑎)
396ad3antrrr 765 . . . . . . . . . . . . . 14 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ 𝑤𝑈) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎) → 𝑎𝑋)
40 simplr 791 . . . . . . . . . . . . . . 15 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ 𝑤𝑈) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎) → 𝑤𝑈)
41 eqid 2621 . . . . . . . . . . . . . . . . . 18 (𝑤 “ {𝑝}) = (𝑤 “ {𝑝})
42 imaeq1 5420 . . . . . . . . . . . . . . . . . . . 20 (𝑢 = 𝑤 → (𝑢 “ {𝑝}) = (𝑤 “ {𝑝}))
4342eqeq2d 2631 . . . . . . . . . . . . . . . . . . 19 (𝑢 = 𝑤 → ((𝑤 “ {𝑝}) = (𝑢 “ {𝑝}) ↔ (𝑤 “ {𝑝}) = (𝑤 “ {𝑝})))
4443rspcev 3295 . . . . . . . . . . . . . . . . . 18 ((𝑤𝑈 ∧ (𝑤 “ {𝑝}) = (𝑤 “ {𝑝})) → ∃𝑢𝑈 (𝑤 “ {𝑝}) = (𝑢 “ {𝑝}))
4541, 44mpan2 706 . . . . . . . . . . . . . . . . 17 (𝑤𝑈 → ∃𝑢𝑈 (𝑤 “ {𝑝}) = (𝑢 “ {𝑝}))
4645adantl 482 . . . . . . . . . . . . . . . 16 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑤𝑈) → ∃𝑢𝑈 (𝑤 “ {𝑝}) = (𝑢 “ {𝑝}))
47 vex 3189 . . . . . . . . . . . . . . . . . . 19 𝑤 ∈ V
4847imaex 7051 . . . . . . . . . . . . . . . . . 18 (𝑤 “ {𝑝}) ∈ V
4913ustuqtoplem 21953 . . . . . . . . . . . . . . . . . 18 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ (𝑤 “ {𝑝}) ∈ V) → ((𝑤 “ {𝑝}) ∈ (𝑁𝑝) ↔ ∃𝑢𝑈 (𝑤 “ {𝑝}) = (𝑢 “ {𝑝})))
5048, 49mpan2 706 . . . . . . . . . . . . . . . . 17 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → ((𝑤 “ {𝑝}) ∈ (𝑁𝑝) ↔ ∃𝑢𝑈 (𝑤 “ {𝑝}) = (𝑢 “ {𝑝})))
5150adantr 481 . . . . . . . . . . . . . . . 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 1322 . . . . . . . . . . . . . 14 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ 𝑤𝑈) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎) → (𝑤 “ {𝑝}) ∈ (𝑁𝑝))
54 sseq1 3605 . . . . . . . . . . . . . . . . . 18 (𝑏 = (𝑤 “ {𝑝}) → (𝑏𝑎 ↔ (𝑤 “ {𝑝}) ⊆ 𝑎))
55543anbi2d 1401 . . . . . . . . . . . . . . . . 17 (𝑏 = (𝑤 “ {𝑝}) → (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑏𝑎𝑎𝑋) ↔ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎𝑎𝑋)))
56 eleq1 2686 . . . . . . . . . . . . . . . . 17 (𝑏 = (𝑤 “ {𝑝}) → (𝑏 ∈ (𝑁𝑝) ↔ (𝑤 “ {𝑝}) ∈ (𝑁𝑝)))
5755, 56anbi12d 746 . . . . . . . . . . . . . . . 16 (𝑏 = (𝑤 “ {𝑝}) → ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑏𝑎𝑎𝑋) ∧ 𝑏 ∈ (𝑁𝑝)) ↔ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎𝑎𝑋) ∧ (𝑤 “ {𝑝}) ∈ (𝑁𝑝))))
5857imbi1d 331 . . . . . . . . . . . . . . 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 21955 . . . . . . . . . . . . . . 15 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑏𝑎𝑎𝑋) ∧ 𝑏 ∈ (𝑁𝑝)) → 𝑎 ∈ (𝑁𝑝))
6048, 58, 59vtocl 3245 . . . . . . . . . . . . . 14 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ (𝑤 “ {𝑝}) ⊆ 𝑎𝑎𝑋) ∧ (𝑤 “ {𝑝}) ∈ (𝑁𝑝)) → 𝑎 ∈ (𝑁𝑝))
6137, 38, 39, 53, 60syl31anc 1326 . . . . . . . . . . . . 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 3070 . . . . . . . . . . 11 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑝𝑎) ∧ ∃𝑤𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎) → ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑝}))
6534, 64impbida 876 . . . . . . . . . 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 2979 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) → (∀𝑝𝑎 𝑎 ∈ (𝑁𝑝) ↔ ∀𝑝𝑎𝑤𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎))
6867rabbidva 3176 . . . . . . 7 (𝑈 ∈ (UnifOn‘𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)} = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎𝑤𝑈 (𝑤 “ {𝑝}) ⊆ 𝑎})
693, 68eqtr4d 2658 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑋) → 𝐽 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)})
70 utopsnneip.1 . . . . . 6 𝐾 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)}
7169, 70syl6eqr 2673 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → 𝐽 = 𝐾)
7271fveq2d 6152 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (nei‘𝐽) = (nei‘𝐾))
7372fveq1d 6150 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → ((nei‘𝐽)‘{𝑃}) = ((nei‘𝐾)‘{𝑃}))
7473adantr 481 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → ((nei‘𝐽)‘{𝑃}) = ((nei‘𝐾)‘{𝑃}))
7513ustuqtop0 21954 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → 𝑁:𝑋⟶𝒫 𝒫 𝑋)
7613ustuqtop1 21955 . . . . 5 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → 𝑏 ∈ (𝑁𝑝))
7713ustuqtop2 21956 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (fi‘(𝑁𝑝)) ⊆ (𝑁𝑝))
7813ustuqtop3 21957 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → 𝑝𝑎)
7913ustuqtop4 21958 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → ∃𝑏 ∈ (𝑁𝑝)∀𝑞𝑏 𝑎 ∈ (𝑁𝑞))
8013ustuqtop5 21959 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → 𝑋 ∈ (𝑁𝑝))
8170, 75, 76, 77, 78, 79, 80neiptopnei 20846 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → 𝑁 = (𝑝𝑋 ↦ ((nei‘𝐾)‘{𝑝})))
8281adantr 481 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → 𝑁 = (𝑝𝑋 ↦ ((nei‘𝐾)‘{𝑝})))
83 simpr 477 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) ∧ 𝑝 = 𝑃) → 𝑝 = 𝑃)
8483sneqd 4160 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) ∧ 𝑝 = 𝑃) → {𝑝} = {𝑃})
8584fveq2d 6152 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) ∧ 𝑝 = 𝑃) → ((nei‘𝐾)‘{𝑝}) = ((nei‘𝐾)‘{𝑃}))
86 simpr 477 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → 𝑃𝑋)
87 fvex 6158 . . . 4 ((nei‘𝐾)‘{𝑃}) ∈ V
8887a1i 11 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → ((nei‘𝐾)‘{𝑃}) ∈ V)
8982, 85, 86, 88fvmptd 6245 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → (𝑁𝑃) = ((nei‘𝐾)‘{𝑃}))
90 mptexg 6438 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
91 rnexg 7045 . . . . 5 ((𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V → ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
9290, 91syl 17 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
9392adantr 481 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
9413a1i 11 . . . 4 ((𝑃𝑋 ∧ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) → 𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝}))))
95 nfv 1840 . . . . . . . 8 𝑣 𝑃𝑋
96 nfmpt1 4707 . . . . . . . . . 10 𝑣(𝑣𝑈 ↦ (𝑣 “ {𝑃}))
9796nfrn 5328 . . . . . . . . 9 𝑣ran (𝑣𝑈 ↦ (𝑣 “ {𝑃}))
9897nfel1 2775 . . . . . . . 8 𝑣ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V
9995, 98nfan 1825 . . . . . . 7 𝑣(𝑃𝑋 ∧ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
100 nfv 1840 . . . . . . 7 𝑣 𝑝 = 𝑃
10199, 100nfan 1825 . . . . . 6 𝑣((𝑃𝑋 ∧ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) ∧ 𝑝 = 𝑃)
102 simpr2 1066 . . . . . . . . 9 ((𝑃𝑋 ∧ (ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V ∧ 𝑝 = 𝑃𝑣𝑈)) → 𝑝 = 𝑃)
103102sneqd 4160 . . . . . . . 8 ((𝑃𝑋 ∧ (ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V ∧ 𝑝 = 𝑃𝑣𝑈)) → {𝑝} = {𝑃})
104103imaeq2d 5425 . . . . . . 7 ((𝑃𝑋 ∧ (ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V ∧ 𝑝 = 𝑃𝑣𝑈)) → (𝑣 “ {𝑝}) = (𝑣 “ {𝑃}))
1051043anassrs 1287 . . . . . 6 ((((𝑃𝑋 ∧ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) ∧ 𝑝 = 𝑃) ∧ 𝑣𝑈) → (𝑣 “ {𝑝}) = (𝑣 “ {𝑃}))
106101, 105mpteq2da 4703 . . . . 5 (((𝑃𝑋 ∧ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) ∧ 𝑝 = 𝑃) → (𝑣𝑈 ↦ (𝑣 “ {𝑝})) = (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
107106rneqd 5313 . . . 4 (((𝑃𝑋 ∧ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) ∧ 𝑝 = 𝑃) → ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
108 simpl 473 . . . 4 ((𝑃𝑋 ∧ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) → 𝑃𝑋)
109 simpr 477 . . . 4 ((𝑃𝑋 ∧ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) → ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
11094, 107, 108, 109fvmptd 6245 . . 3 ((𝑃𝑋 ∧ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V) → (𝑁𝑃) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
11186, 93, 110syl2anc 692 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → (𝑁𝑃) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
11274, 89, 1113eqtr2d 2661 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 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  wrex 2908  {crab 2911  Vcvv 3186  wss 3555  𝒫 cpw 4130  {csn 4148  cmpt 4673  ran crn 5075  cima 5077  cfv 5847  neicnei 20811  UnifOncust 21913  unifTopcutop 21944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-en 7900  df-fin 7903  df-fi 8261  df-top 20621  df-nei 20812  df-ust 21914  df-utop 21945
This theorem is referenced by:  utopsnneip  21962
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