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Theorem utopsnneiplem 23752
Description: The neighborhoods of a point 𝑃 for the topology induced by an uniform space π‘ˆ. (Contributed by Thierry Arnoux, 11-Jan-2018.)
Hypotheses
Ref Expression
utoptop.1 𝐽 = (unifTopβ€˜π‘ˆ)
utopsnneip.1 𝐾 = {π‘Ž ∈ 𝒫 𝑋 ∣ βˆ€π‘ ∈ π‘Ž π‘Ž ∈ (π‘β€˜π‘)}
utopsnneip.2 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑝})))
Assertion
Ref Expression
utopsnneiplem ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) β†’ ((neiβ€˜π½)β€˜{𝑃}) = ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑃})))
Distinct variable groups:   𝑝,π‘Ž,𝐾   𝑁,π‘Ž,𝑝   𝑣,𝑝,𝑃   𝑣,π‘Ž,π‘ˆ,𝑝   𝑋,π‘Ž,𝑝,𝑣
Allowed substitution hints:   𝑃(π‘Ž)   𝐽(𝑣,𝑝,π‘Ž)   𝐾(𝑣)   𝑁(𝑣)

Proof of Theorem utopsnneiplem
Dummy variables 𝑏 π‘ž 𝑒 𝑀 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 utoptop.1 . . . . . . . 8 𝐽 = (unifTopβ€˜π‘ˆ)
2 utopval 23737 . . . . . . . 8 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ (unifTopβ€˜π‘ˆ) = {π‘Ž ∈ 𝒫 𝑋 ∣ βˆ€π‘ ∈ π‘Ž βˆƒπ‘€ ∈ π‘ˆ (𝑀 β€œ {𝑝}) βŠ† π‘Ž})
31, 2eqtrid 2785 . . . . . . 7 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ 𝐽 = {π‘Ž ∈ 𝒫 𝑋 ∣ βˆ€π‘ ∈ π‘Ž βˆƒπ‘€ ∈ π‘ˆ (𝑀 β€œ {𝑝}) βŠ† π‘Ž})
4 simpll 766 . . . . . . . . . . 11 (((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ π‘Ž ∈ 𝒫 𝑋) ∧ 𝑝 ∈ π‘Ž) β†’ π‘ˆ ∈ (UnifOnβ€˜π‘‹))
5 simpr 486 . . . . . . . . . . . . 13 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ π‘Ž ∈ 𝒫 𝑋) β†’ π‘Ž ∈ 𝒫 𝑋)
65elpwid 4612 . . . . . . . . . . . 12 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ π‘Ž ∈ 𝒫 𝑋) β†’ π‘Ž βŠ† 𝑋)
76sselda 3983 . . . . . . . . . . 11 (((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ π‘Ž ∈ 𝒫 𝑋) ∧ 𝑝 ∈ π‘Ž) β†’ 𝑝 ∈ 𝑋)
8 simpr 486 . . . . . . . . . . . . . 14 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) β†’ 𝑝 ∈ 𝑋)
9 mptexg 7223 . . . . . . . . . . . . . . . 16 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑝})) ∈ V)
10 rnexg 7895 . . . . . . . . . . . . . . . 16 ((𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑝})) ∈ V β†’ ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑝})) ∈ V)
119, 10syl 17 . . . . . . . . . . . . . . 15 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑝})) ∈ V)
1211adantr 482 . . . . . . . . . . . . . 14 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) β†’ ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑝})) ∈ V)
13 utopsnneip.2 . . . . . . . . . . . . . . 15 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑝})))
1413fvmpt2 7010 . . . . . . . . . . . . . 14 ((𝑝 ∈ 𝑋 ∧ ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑝})) ∈ V) β†’ (π‘β€˜π‘) = ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑝})))
158, 12, 14syl2anc 585 . . . . . . . . . . . . 13 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) β†’ (π‘β€˜π‘) = ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑝})))
1615eleq2d 2820 . . . . . . . . . . . 12 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) β†’ (π‘Ž ∈ (π‘β€˜π‘) ↔ π‘Ž ∈ ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑝}))))
17 eqid 2733 . . . . . . . . . . . . . 14 (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑝})) = (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑝}))
1817elrnmpt 5956 . . . . . . . . . . . . 13 (π‘Ž ∈ V β†’ (π‘Ž ∈ ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑝})) ↔ βˆƒπ‘£ ∈ π‘ˆ π‘Ž = (𝑣 β€œ {𝑝})))
1918elv 3481 . . . . . . . . . . . 12 (π‘Ž ∈ ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑝})) ↔ βˆƒπ‘£ ∈ π‘ˆ π‘Ž = (𝑣 β€œ {𝑝}))
2016, 19bitrdi 287 . . . . . . . . . . 11 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) β†’ (π‘Ž ∈ (π‘β€˜π‘) ↔ βˆƒπ‘£ ∈ π‘ˆ π‘Ž = (𝑣 β€œ {𝑝})))
214, 7, 20syl2anc 585 . . . . . . . . . 10 (((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ π‘Ž ∈ 𝒫 𝑋) ∧ 𝑝 ∈ π‘Ž) β†’ (π‘Ž ∈ (π‘β€˜π‘) ↔ βˆƒπ‘£ ∈ π‘ˆ π‘Ž = (𝑣 β€œ {𝑝})))
22 nfv 1918 . . . . . . . . . . . . 13 Ⅎ𝑣((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ π‘Ž ∈ 𝒫 𝑋) ∧ 𝑝 ∈ π‘Ž)
23 nfre1 3283 . . . . . . . . . . . . 13 β„²π‘£βˆƒπ‘£ ∈ π‘ˆ π‘Ž = (𝑣 β€œ {𝑝})
2422, 23nfan 1903 . . . . . . . . . . . 12 Ⅎ𝑣(((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ π‘Ž ∈ 𝒫 𝑋) ∧ 𝑝 ∈ π‘Ž) ∧ βˆƒπ‘£ ∈ π‘ˆ π‘Ž = (𝑣 β€œ {𝑝}))
25 simplr 768 . . . . . . . . . . . . 13 ((((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ π‘Ž ∈ 𝒫 𝑋) ∧ 𝑝 ∈ π‘Ž) ∧ βˆƒπ‘£ ∈ π‘ˆ π‘Ž = (𝑣 β€œ {𝑝})) ∧ 𝑣 ∈ π‘ˆ) ∧ π‘Ž = (𝑣 β€œ {𝑝})) β†’ 𝑣 ∈ π‘ˆ)
26 eqimss2 4042 . . . . . . . . . . . . . 14 (π‘Ž = (𝑣 β€œ {𝑝}) β†’ (𝑣 β€œ {𝑝}) βŠ† π‘Ž)
2726adantl 483 . . . . . . . . . . . . 13 ((((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ π‘Ž ∈ 𝒫 𝑋) ∧ 𝑝 ∈ π‘Ž) ∧ βˆƒπ‘£ ∈ π‘ˆ π‘Ž = (𝑣 β€œ {𝑝})) ∧ 𝑣 ∈ π‘ˆ) ∧ π‘Ž = (𝑣 β€œ {𝑝})) β†’ (𝑣 β€œ {𝑝}) βŠ† π‘Ž)
28 imaeq1 6055 . . . . . . . . . . . . . . 15 (𝑀 = 𝑣 β†’ (𝑀 β€œ {𝑝}) = (𝑣 β€œ {𝑝}))
2928sseq1d 4014 . . . . . . . . . . . . . 14 (𝑀 = 𝑣 β†’ ((𝑀 β€œ {𝑝}) βŠ† π‘Ž ↔ (𝑣 β€œ {𝑝}) βŠ† π‘Ž))
3029rspcev 3613 . . . . . . . . . . . . 13 ((𝑣 ∈ π‘ˆ ∧ (𝑣 β€œ {𝑝}) βŠ† π‘Ž) β†’ βˆƒπ‘€ ∈ π‘ˆ (𝑀 β€œ {𝑝}) βŠ† π‘Ž)
3125, 27, 30syl2anc 585 . . . . . . . . . . . 12 ((((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ π‘Ž ∈ 𝒫 𝑋) ∧ 𝑝 ∈ π‘Ž) ∧ βˆƒπ‘£ ∈ π‘ˆ π‘Ž = (𝑣 β€œ {𝑝})) ∧ 𝑣 ∈ π‘ˆ) ∧ π‘Ž = (𝑣 β€œ {𝑝})) β†’ βˆƒπ‘€ ∈ π‘ˆ (𝑀 β€œ {𝑝}) βŠ† π‘Ž)
32 simpr 486 . . . . . . . . . . . 12 ((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ π‘Ž ∈ 𝒫 𝑋) ∧ 𝑝 ∈ π‘Ž) ∧ βˆƒπ‘£ ∈ π‘ˆ π‘Ž = (𝑣 β€œ {𝑝})) β†’ βˆƒπ‘£ ∈ π‘ˆ π‘Ž = (𝑣 β€œ {𝑝}))
3324, 31, 32r19.29af 3266 . . . . . . . . . . 11 ((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ π‘Ž ∈ 𝒫 𝑋) ∧ 𝑝 ∈ π‘Ž) ∧ βˆƒπ‘£ ∈ π‘ˆ π‘Ž = (𝑣 β€œ {𝑝})) β†’ βˆƒπ‘€ ∈ π‘ˆ (𝑀 β€œ {𝑝}) βŠ† π‘Ž)
344ad2antrr 725 . . . . . . . . . . . . . . 15 (((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ π‘Ž ∈ 𝒫 𝑋) ∧ 𝑝 ∈ π‘Ž) ∧ 𝑀 ∈ π‘ˆ) ∧ (𝑀 β€œ {𝑝}) βŠ† π‘Ž) β†’ π‘ˆ ∈ (UnifOnβ€˜π‘‹))
357ad2antrr 725 . . . . . . . . . . . . . . 15 (((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ π‘Ž ∈ 𝒫 𝑋) ∧ 𝑝 ∈ π‘Ž) ∧ 𝑀 ∈ π‘ˆ) ∧ (𝑀 β€œ {𝑝}) βŠ† π‘Ž) β†’ 𝑝 ∈ 𝑋)
3634, 35jca 513 . . . . . . . . . . . . . 14 (((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ π‘Ž ∈ 𝒫 𝑋) ∧ 𝑝 ∈ π‘Ž) ∧ 𝑀 ∈ π‘ˆ) ∧ (𝑀 β€œ {𝑝}) βŠ† π‘Ž) β†’ (π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋))
37 simpr 486 . . . . . . . . . . . . . 14 (((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ π‘Ž ∈ 𝒫 𝑋) ∧ 𝑝 ∈ π‘Ž) ∧ 𝑀 ∈ π‘ˆ) ∧ (𝑀 β€œ {𝑝}) βŠ† π‘Ž) β†’ (𝑀 β€œ {𝑝}) βŠ† π‘Ž)
386ad3antrrr 729 . . . . . . . . . . . . . 14 (((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ π‘Ž ∈ 𝒫 𝑋) ∧ 𝑝 ∈ π‘Ž) ∧ 𝑀 ∈ π‘ˆ) ∧ (𝑀 β€œ {𝑝}) βŠ† π‘Ž) β†’ π‘Ž βŠ† 𝑋)
39 simplr 768 . . . . . . . . . . . . . . 15 (((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ π‘Ž ∈ 𝒫 𝑋) ∧ 𝑝 ∈ π‘Ž) ∧ 𝑀 ∈ π‘ˆ) ∧ (𝑀 β€œ {𝑝}) βŠ† π‘Ž) β†’ 𝑀 ∈ π‘ˆ)
40 eqid 2733 . . . . . . . . . . . . . . . . . 18 (𝑀 β€œ {𝑝}) = (𝑀 β€œ {𝑝})
41 imaeq1 6055 . . . . . . . . . . . . . . . . . . 19 (𝑒 = 𝑀 β†’ (𝑒 β€œ {𝑝}) = (𝑀 β€œ {𝑝}))
4241rspceeqv 3634 . . . . . . . . . . . . . . . . . 18 ((𝑀 ∈ π‘ˆ ∧ (𝑀 β€œ {𝑝}) = (𝑀 β€œ {𝑝})) β†’ βˆƒπ‘’ ∈ π‘ˆ (𝑀 β€œ {𝑝}) = (𝑒 β€œ {𝑝}))
4340, 42mpan2 690 . . . . . . . . . . . . . . . . 17 (𝑀 ∈ π‘ˆ β†’ βˆƒπ‘’ ∈ π‘ˆ (𝑀 β€œ {𝑝}) = (𝑒 β€œ {𝑝}))
4443adantl 483 . . . . . . . . . . . . . . . 16 (((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ 𝑀 ∈ π‘ˆ) β†’ βˆƒπ‘’ ∈ π‘ˆ (𝑀 β€œ {𝑝}) = (𝑒 β€œ {𝑝}))
45 vex 3479 . . . . . . . . . . . . . . . . . . 19 𝑀 ∈ V
4645imaex 7907 . . . . . . . . . . . . . . . . . 18 (𝑀 β€œ {𝑝}) ∈ V
4713ustuqtoplem 23744 . . . . . . . . . . . . . . . . . 18 (((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ (𝑀 β€œ {𝑝}) ∈ V) β†’ ((𝑀 β€œ {𝑝}) ∈ (π‘β€˜π‘) ↔ βˆƒπ‘’ ∈ π‘ˆ (𝑀 β€œ {𝑝}) = (𝑒 β€œ {𝑝})))
4846, 47mpan2 690 . . . . . . . . . . . . . . . . 17 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) β†’ ((𝑀 β€œ {𝑝}) ∈ (π‘β€˜π‘) ↔ βˆƒπ‘’ ∈ π‘ˆ (𝑀 β€œ {𝑝}) = (𝑒 β€œ {𝑝})))
4948adantr 482 . . . . . . . . . . . . . . . 16 (((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ 𝑀 ∈ π‘ˆ) β†’ ((𝑀 β€œ {𝑝}) ∈ (π‘β€˜π‘) ↔ βˆƒπ‘’ ∈ π‘ˆ (𝑀 β€œ {𝑝}) = (𝑒 β€œ {𝑝})))
5044, 49mpbird 257 . . . . . . . . . . . . . . 15 (((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ 𝑀 ∈ π‘ˆ) β†’ (𝑀 β€œ {𝑝}) ∈ (π‘β€˜π‘))
5134, 35, 39, 50syl21anc 837 . . . . . . . . . . . . . 14 (((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ π‘Ž ∈ 𝒫 𝑋) ∧ 𝑝 ∈ π‘Ž) ∧ 𝑀 ∈ π‘ˆ) ∧ (𝑀 β€œ {𝑝}) βŠ† π‘Ž) β†’ (𝑀 β€œ {𝑝}) ∈ (π‘β€˜π‘))
52 sseq1 4008 . . . . . . . . . . . . . . . . . 18 (𝑏 = (𝑀 β€œ {𝑝}) β†’ (𝑏 βŠ† π‘Ž ↔ (𝑀 β€œ {𝑝}) βŠ† π‘Ž))
53523anbi2d 1442 . . . . . . . . . . . . . . . . 17 (𝑏 = (𝑀 β€œ {𝑝}) β†’ (((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ 𝑏 βŠ† π‘Ž ∧ π‘Ž βŠ† 𝑋) ↔ ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ (𝑀 β€œ {𝑝}) βŠ† π‘Ž ∧ π‘Ž βŠ† 𝑋)))
54 eleq1 2822 . . . . . . . . . . . . . . . . 17 (𝑏 = (𝑀 β€œ {𝑝}) β†’ (𝑏 ∈ (π‘β€˜π‘) ↔ (𝑀 β€œ {𝑝}) ∈ (π‘β€˜π‘)))
5553, 54anbi12d 632 . . . . . . . . . . . . . . . 16 (𝑏 = (𝑀 β€œ {𝑝}) β†’ ((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ 𝑏 βŠ† π‘Ž ∧ π‘Ž βŠ† 𝑋) ∧ 𝑏 ∈ (π‘β€˜π‘)) ↔ (((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ (𝑀 β€œ {𝑝}) βŠ† π‘Ž ∧ π‘Ž βŠ† 𝑋) ∧ (𝑀 β€œ {𝑝}) ∈ (π‘β€˜π‘))))
5655imbi1d 342 . . . . . . . . . . . . . . 15 (𝑏 = (𝑀 β€œ {𝑝}) β†’ (((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ 𝑏 βŠ† π‘Ž ∧ π‘Ž βŠ† 𝑋) ∧ 𝑏 ∈ (π‘β€˜π‘)) β†’ π‘Ž ∈ (π‘β€˜π‘)) ↔ ((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ (𝑀 β€œ {𝑝}) βŠ† π‘Ž ∧ π‘Ž βŠ† 𝑋) ∧ (𝑀 β€œ {𝑝}) ∈ (π‘β€˜π‘)) β†’ π‘Ž ∈ (π‘β€˜π‘))))
5713ustuqtop1 23746 . . . . . . . . . . . . . . 15 ((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ 𝑏 βŠ† π‘Ž ∧ π‘Ž βŠ† 𝑋) ∧ 𝑏 ∈ (π‘β€˜π‘)) β†’ π‘Ž ∈ (π‘β€˜π‘))
5846, 56, 57vtocl 3550 . . . . . . . . . . . . . 14 ((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ (𝑀 β€œ {𝑝}) βŠ† π‘Ž ∧ π‘Ž βŠ† 𝑋) ∧ (𝑀 β€œ {𝑝}) ∈ (π‘β€˜π‘)) β†’ π‘Ž ∈ (π‘β€˜π‘))
5936, 37, 38, 51, 58syl31anc 1374 . . . . . . . . . . . . 13 (((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ π‘Ž ∈ 𝒫 𝑋) ∧ 𝑝 ∈ π‘Ž) ∧ 𝑀 ∈ π‘ˆ) ∧ (𝑀 β€œ {𝑝}) βŠ† π‘Ž) β†’ π‘Ž ∈ (π‘β€˜π‘))
6036, 20syl 17 . . . . . . . . . . . . 13 (((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ π‘Ž ∈ 𝒫 𝑋) ∧ 𝑝 ∈ π‘Ž) ∧ 𝑀 ∈ π‘ˆ) ∧ (𝑀 β€œ {𝑝}) βŠ† π‘Ž) β†’ (π‘Ž ∈ (π‘β€˜π‘) ↔ βˆƒπ‘£ ∈ π‘ˆ π‘Ž = (𝑣 β€œ {𝑝})))
6159, 60mpbid 231 . . . . . . . . . . . 12 (((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ π‘Ž ∈ 𝒫 𝑋) ∧ 𝑝 ∈ π‘Ž) ∧ 𝑀 ∈ π‘ˆ) ∧ (𝑀 β€œ {𝑝}) βŠ† π‘Ž) β†’ βˆƒπ‘£ ∈ π‘ˆ π‘Ž = (𝑣 β€œ {𝑝}))
6261r19.29an 3159 . . . . . . . . . . 11 ((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ π‘Ž ∈ 𝒫 𝑋) ∧ 𝑝 ∈ π‘Ž) ∧ βˆƒπ‘€ ∈ π‘ˆ (𝑀 β€œ {𝑝}) βŠ† π‘Ž) β†’ βˆƒπ‘£ ∈ π‘ˆ π‘Ž = (𝑣 β€œ {𝑝}))
6333, 62impbida 800 . . . . . . . . . 10 (((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ π‘Ž ∈ 𝒫 𝑋) ∧ 𝑝 ∈ π‘Ž) β†’ (βˆƒπ‘£ ∈ π‘ˆ π‘Ž = (𝑣 β€œ {𝑝}) ↔ βˆƒπ‘€ ∈ π‘ˆ (𝑀 β€œ {𝑝}) βŠ† π‘Ž))
6421, 63bitrd 279 . . . . . . . . 9 (((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ π‘Ž ∈ 𝒫 𝑋) ∧ 𝑝 ∈ π‘Ž) β†’ (π‘Ž ∈ (π‘β€˜π‘) ↔ βˆƒπ‘€ ∈ π‘ˆ (𝑀 β€œ {𝑝}) βŠ† π‘Ž))
6564ralbidva 3176 . . . . . . . 8 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ π‘Ž ∈ 𝒫 𝑋) β†’ (βˆ€π‘ ∈ π‘Ž π‘Ž ∈ (π‘β€˜π‘) ↔ βˆ€π‘ ∈ π‘Ž βˆƒπ‘€ ∈ π‘ˆ (𝑀 β€œ {𝑝}) βŠ† π‘Ž))
6665rabbidva 3440 . . . . . . 7 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ {π‘Ž ∈ 𝒫 𝑋 ∣ βˆ€π‘ ∈ π‘Ž π‘Ž ∈ (π‘β€˜π‘)} = {π‘Ž ∈ 𝒫 𝑋 ∣ βˆ€π‘ ∈ π‘Ž βˆƒπ‘€ ∈ π‘ˆ (𝑀 β€œ {𝑝}) βŠ† π‘Ž})
673, 66eqtr4d 2776 . . . . . 6 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ 𝐽 = {π‘Ž ∈ 𝒫 𝑋 ∣ βˆ€π‘ ∈ π‘Ž π‘Ž ∈ (π‘β€˜π‘)})
68 utopsnneip.1 . . . . . 6 𝐾 = {π‘Ž ∈ 𝒫 𝑋 ∣ βˆ€π‘ ∈ π‘Ž π‘Ž ∈ (π‘β€˜π‘)}
6967, 68eqtr4di 2791 . . . . 5 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ 𝐽 = 𝐾)
7069fveq2d 6896 . . . 4 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ (neiβ€˜π½) = (neiβ€˜πΎ))
7170fveq1d 6894 . . 3 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ ((neiβ€˜π½)β€˜{𝑃}) = ((neiβ€˜πΎ)β€˜{𝑃}))
7271adantr 482 . 2 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) β†’ ((neiβ€˜π½)β€˜{𝑃}) = ((neiβ€˜πΎ)β€˜{𝑃}))
7313ustuqtop0 23745 . . . . 5 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ 𝑁:π‘‹βŸΆπ’« 𝒫 𝑋)
7413ustuqtop1 23746 . . . . 5 ((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž βŠ† 𝑏 ∧ 𝑏 βŠ† 𝑋) ∧ π‘Ž ∈ (π‘β€˜π‘)) β†’ 𝑏 ∈ (π‘β€˜π‘))
7513ustuqtop2 23747 . . . . 5 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) β†’ (fiβ€˜(π‘β€˜π‘)) βŠ† (π‘β€˜π‘))
7613ustuqtop3 23748 . . . . 5 (((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž ∈ (π‘β€˜π‘)) β†’ 𝑝 ∈ π‘Ž)
7713ustuqtop4 23749 . . . . 5 (((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž ∈ (π‘β€˜π‘)) β†’ βˆƒπ‘ ∈ (π‘β€˜π‘)βˆ€π‘ž ∈ 𝑏 π‘Ž ∈ (π‘β€˜π‘ž))
7813ustuqtop5 23750 . . . . 5 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) β†’ 𝑋 ∈ (π‘β€˜π‘))
7968, 73, 74, 75, 76, 77, 78neiptopnei 22636 . . . 4 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((neiβ€˜πΎ)β€˜{𝑝})))
8079adantr 482 . . 3 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) β†’ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((neiβ€˜πΎ)β€˜{𝑝})))
81 simpr 486 . . . . 5 (((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) ∧ 𝑝 = 𝑃) β†’ 𝑝 = 𝑃)
8281sneqd 4641 . . . 4 (((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) ∧ 𝑝 = 𝑃) β†’ {𝑝} = {𝑃})
8382fveq2d 6896 . . 3 (((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) ∧ 𝑝 = 𝑃) β†’ ((neiβ€˜πΎ)β€˜{𝑝}) = ((neiβ€˜πΎ)β€˜{𝑃}))
84 simpr 486 . . 3 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) β†’ 𝑃 ∈ 𝑋)
85 fvexd 6907 . . 3 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) β†’ ((neiβ€˜πΎ)β€˜{𝑃}) ∈ V)
8680, 83, 84, 85fvmptd 7006 . 2 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) β†’ (π‘β€˜π‘ƒ) = ((neiβ€˜πΎ)β€˜{𝑃}))
87 mptexg 7223 . . . . 5 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑃})) ∈ V)
88 rnexg 7895 . . . . 5 ((𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑃})) ∈ V β†’ ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑃})) ∈ V)
8987, 88syl 17 . . . 4 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑃})) ∈ V)
9089adantr 482 . . 3 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) β†’ ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑃})) ∈ V)
91 nfv 1918 . . . . . . . 8 Ⅎ𝑣 𝑃 ∈ 𝑋
92 nfmpt1 5257 . . . . . . . . . 10 Ⅎ𝑣(𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑃}))
9392nfrn 5952 . . . . . . . . 9 Ⅎ𝑣ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑃}))
9493nfel1 2920 . . . . . . . 8 Ⅎ𝑣ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑃})) ∈ V
9591, 94nfan 1903 . . . . . . 7 Ⅎ𝑣(𝑃 ∈ 𝑋 ∧ ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑃})) ∈ V)
96 nfv 1918 . . . . . . 7 Ⅎ𝑣 𝑝 = 𝑃
9795, 96nfan 1903 . . . . . 6 Ⅎ𝑣((𝑃 ∈ 𝑋 ∧ ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑃})) ∈ V) ∧ 𝑝 = 𝑃)
98 simpr2 1196 . . . . . . . . 9 ((𝑃 ∈ 𝑋 ∧ (ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑃})) ∈ V ∧ 𝑝 = 𝑃 ∧ 𝑣 ∈ π‘ˆ)) β†’ 𝑝 = 𝑃)
9998sneqd 4641 . . . . . . . 8 ((𝑃 ∈ 𝑋 ∧ (ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑃})) ∈ V ∧ 𝑝 = 𝑃 ∧ 𝑣 ∈ π‘ˆ)) β†’ {𝑝} = {𝑃})
10099imaeq2d 6060 . . . . . . 7 ((𝑃 ∈ 𝑋 ∧ (ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑃})) ∈ V ∧ 𝑝 = 𝑃 ∧ 𝑣 ∈ π‘ˆ)) β†’ (𝑣 β€œ {𝑝}) = (𝑣 β€œ {𝑃}))
1011003anassrs 1361 . . . . . 6 ((((𝑃 ∈ 𝑋 ∧ ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑃})) ∈ V) ∧ 𝑝 = 𝑃) ∧ 𝑣 ∈ π‘ˆ) β†’ (𝑣 β€œ {𝑝}) = (𝑣 β€œ {𝑃}))
10297, 101mpteq2da 5247 . . . . 5 (((𝑃 ∈ 𝑋 ∧ ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑃})) ∈ V) ∧ 𝑝 = 𝑃) β†’ (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑝})) = (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑃})))
103102rneqd 5938 . . . 4 (((𝑃 ∈ 𝑋 ∧ ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑃})) ∈ V) ∧ 𝑝 = 𝑃) β†’ ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑝})) = ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑃})))
104 simpl 484 . . . 4 ((𝑃 ∈ 𝑋 ∧ ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑃})) ∈ V) β†’ 𝑃 ∈ 𝑋)
105 simpr 486 . . . 4 ((𝑃 ∈ 𝑋 ∧ ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑃})) ∈ V) β†’ ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑃})) ∈ V)
10613, 103, 104, 105fvmptd2 7007 . . 3 ((𝑃 ∈ 𝑋 ∧ ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑃})) ∈ V) β†’ (π‘β€˜π‘ƒ) = ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑃})))
10784, 90, 106syl2anc 585 . 2 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) β†’ (π‘β€˜π‘ƒ) = ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑃})))
10872, 86, 1073eqtr2d 2779 1 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) β†’ ((neiβ€˜π½)β€˜{𝑃}) = ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑃})))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  βˆƒwrex 3071  {crab 3433  Vcvv 3475   βŠ† wss 3949  π’« cpw 4603  {csn 4629   ↦ cmpt 5232  ran crn 5678   β€œ cima 5680  β€˜cfv 6544  neicnei 22601  UnifOncust 23704  unifTopcutop 23735
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-om 7856  df-1o 8466  df-er 8703  df-en 8940  df-fin 8943  df-fi 9406  df-top 22396  df-nei 22602  df-ust 23705  df-utop 23736
This theorem is referenced by:  utopsnneip  23753
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