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Theorem neiptopreu 22857
Description: If, to each element 𝑃 of a set 𝑋, we associate a set (π‘β€˜π‘ƒ) fulfilling Properties Vi, Vii, Viii and Property Viv of [BourbakiTop1] p. I.2. , corresponding to ssnei 22834, innei 22849, elnei 22835 and neissex 22851, then there is a unique topology 𝑗 such that for any point 𝑝, (π‘β€˜π‘) is the set of neighborhoods of 𝑝. Proposition 2 of [BourbakiTop1] p. I.3. This can be used to build a topology from a set of neighborhoods. Note that innei 22849 uses binary intersections whereas Property Vii mentions finite intersections (which includes the empty intersection of subsets of 𝑋, which is equal to 𝑋), so we add the hypothesis that 𝑋 is a neighborhood of all points. TODO: when df-fi 9408 includes the empty intersection, remove that extra hypothesis. (Contributed by Thierry Arnoux, 6-Jan-2018.)
Hypotheses
Ref Expression
neiptop.o 𝐽 = {π‘Ž ∈ 𝒫 𝑋 ∣ βˆ€π‘ ∈ π‘Ž π‘Ž ∈ (π‘β€˜π‘)}
neiptop.0 (πœ‘ β†’ 𝑁:π‘‹βŸΆπ’« 𝒫 𝑋)
neiptop.1 ((((πœ‘ ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž βŠ† 𝑏 ∧ 𝑏 βŠ† 𝑋) ∧ π‘Ž ∈ (π‘β€˜π‘)) β†’ 𝑏 ∈ (π‘β€˜π‘))
neiptop.2 ((πœ‘ ∧ 𝑝 ∈ 𝑋) β†’ (fiβ€˜(π‘β€˜π‘)) βŠ† (π‘β€˜π‘))
neiptop.3 (((πœ‘ ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž ∈ (π‘β€˜π‘)) β†’ 𝑝 ∈ π‘Ž)
neiptop.4 (((πœ‘ ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž ∈ (π‘β€˜π‘)) β†’ βˆƒπ‘ ∈ (π‘β€˜π‘)βˆ€π‘ž ∈ 𝑏 π‘Ž ∈ (π‘β€˜π‘ž))
neiptop.5 ((πœ‘ ∧ 𝑝 ∈ 𝑋) β†’ 𝑋 ∈ (π‘β€˜π‘))
Assertion
Ref Expression
neiptopreu (πœ‘ β†’ βˆƒ!𝑗 ∈ (TopOnβ€˜π‘‹)𝑁 = (𝑝 ∈ 𝑋 ↦ ((neiβ€˜π‘—)β€˜{𝑝})))
Distinct variable groups:   𝑝,π‘Ž,𝑁   𝑋,π‘Ž,𝑏,𝑝   𝐽,π‘Ž,𝑝   𝑋,𝑝   πœ‘,𝑝   𝑁,𝑏   𝑋,𝑏   πœ‘,π‘Ž,𝑏,π‘ž,𝑝   𝑁,𝑝,π‘ž   𝑋,π‘ž   πœ‘,π‘ž   𝑗,π‘Ž,𝑏,𝐽,𝑝   𝑗,π‘ž,𝑁   𝑗,𝑋   πœ‘,𝑗
Allowed substitution hint:   𝐽(π‘ž)

Proof of Theorem neiptopreu
StepHypRef Expression
1 neiptop.o . . . . 5 𝐽 = {π‘Ž ∈ 𝒫 𝑋 ∣ βˆ€π‘ ∈ π‘Ž π‘Ž ∈ (π‘β€˜π‘)}
2 neiptop.0 . . . . 5 (πœ‘ β†’ 𝑁:π‘‹βŸΆπ’« 𝒫 𝑋)
3 neiptop.1 . . . . 5 ((((πœ‘ ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž βŠ† 𝑏 ∧ 𝑏 βŠ† 𝑋) ∧ π‘Ž ∈ (π‘β€˜π‘)) β†’ 𝑏 ∈ (π‘β€˜π‘))
4 neiptop.2 . . . . 5 ((πœ‘ ∧ 𝑝 ∈ 𝑋) β†’ (fiβ€˜(π‘β€˜π‘)) βŠ† (π‘β€˜π‘))
5 neiptop.3 . . . . 5 (((πœ‘ ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž ∈ (π‘β€˜π‘)) β†’ 𝑝 ∈ π‘Ž)
6 neiptop.4 . . . . 5 (((πœ‘ ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž ∈ (π‘β€˜π‘)) β†’ βˆƒπ‘ ∈ (π‘β€˜π‘)βˆ€π‘ž ∈ 𝑏 π‘Ž ∈ (π‘β€˜π‘ž))
7 neiptop.5 . . . . 5 ((πœ‘ ∧ 𝑝 ∈ 𝑋) β†’ 𝑋 ∈ (π‘β€˜π‘))
81, 2, 3, 4, 5, 6, 7neiptoptop 22855 . . . 4 (πœ‘ β†’ 𝐽 ∈ Top)
9 toptopon2 22640 . . . 4 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽))
108, 9sylib 217 . . 3 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽))
111, 2, 3, 4, 5, 6, 7neiptopuni 22854 . . . 4 (πœ‘ β†’ 𝑋 = βˆͺ 𝐽)
1211fveq2d 6894 . . 3 (πœ‘ β†’ (TopOnβ€˜π‘‹) = (TopOnβ€˜βˆͺ 𝐽))
1310, 12eleqtrrd 2834 . 2 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
141, 2, 3, 4, 5, 6, 7neiptopnei 22856 . 2 (πœ‘ β†’ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((neiβ€˜π½)β€˜{𝑝})))
15 nfv 1915 . . . . . . . . . 10 Ⅎ𝑝(πœ‘ ∧ 𝑗 ∈ (TopOnβ€˜π‘‹))
16 nfmpt1 5255 . . . . . . . . . . 11 Ⅎ𝑝(𝑝 ∈ 𝑋 ↦ ((neiβ€˜π‘—)β€˜{𝑝}))
1716nfeq2 2918 . . . . . . . . . 10 Ⅎ𝑝 𝑁 = (𝑝 ∈ 𝑋 ↦ ((neiβ€˜π‘—)β€˜{𝑝}))
1815, 17nfan 1900 . . . . . . . . 9 Ⅎ𝑝((πœ‘ ∧ 𝑗 ∈ (TopOnβ€˜π‘‹)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((neiβ€˜π‘—)β€˜{𝑝})))
19 nfv 1915 . . . . . . . . 9 Ⅎ𝑝 𝑏 βŠ† 𝑋
2018, 19nfan 1900 . . . . . . . 8 Ⅎ𝑝(((πœ‘ ∧ 𝑗 ∈ (TopOnβ€˜π‘‹)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((neiβ€˜π‘—)β€˜{𝑝}))) ∧ 𝑏 βŠ† 𝑋)
21 simpllr 772 . . . . . . . . . . 11 (((((πœ‘ ∧ 𝑗 ∈ (TopOnβ€˜π‘‹)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((neiβ€˜π‘—)β€˜{𝑝}))) ∧ 𝑏 βŠ† 𝑋) ∧ 𝑝 ∈ 𝑏) β†’ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((neiβ€˜π‘—)β€˜{𝑝})))
22 simpr 483 . . . . . . . . . . . 12 ((((πœ‘ ∧ 𝑗 ∈ (TopOnβ€˜π‘‹)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((neiβ€˜π‘—)β€˜{𝑝}))) ∧ 𝑏 βŠ† 𝑋) β†’ 𝑏 βŠ† 𝑋)
2322sselda 3981 . . . . . . . . . . 11 (((((πœ‘ ∧ 𝑗 ∈ (TopOnβ€˜π‘‹)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((neiβ€˜π‘—)β€˜{𝑝}))) ∧ 𝑏 βŠ† 𝑋) ∧ 𝑝 ∈ 𝑏) β†’ 𝑝 ∈ 𝑋)
24 id 22 . . . . . . . . . . . 12 (𝑁 = (𝑝 ∈ 𝑋 ↦ ((neiβ€˜π‘—)β€˜{𝑝})) β†’ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((neiβ€˜π‘—)β€˜{𝑝})))
25 fvexd 6905 . . . . . . . . . . . 12 ((𝑁 = (𝑝 ∈ 𝑋 ↦ ((neiβ€˜π‘—)β€˜{𝑝})) ∧ 𝑝 ∈ 𝑋) β†’ ((neiβ€˜π‘—)β€˜{𝑝}) ∈ V)
2624, 25fvmpt2d 7010 . . . . . . . . . . 11 ((𝑁 = (𝑝 ∈ 𝑋 ↦ ((neiβ€˜π‘—)β€˜{𝑝})) ∧ 𝑝 ∈ 𝑋) β†’ (π‘β€˜π‘) = ((neiβ€˜π‘—)β€˜{𝑝}))
2721, 23, 26syl2anc 582 . . . . . . . . . 10 (((((πœ‘ ∧ 𝑗 ∈ (TopOnβ€˜π‘‹)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((neiβ€˜π‘—)β€˜{𝑝}))) ∧ 𝑏 βŠ† 𝑋) ∧ 𝑝 ∈ 𝑏) β†’ (π‘β€˜π‘) = ((neiβ€˜π‘—)β€˜{𝑝}))
2827eqcomd 2736 . . . . . . . . 9 (((((πœ‘ ∧ 𝑗 ∈ (TopOnβ€˜π‘‹)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((neiβ€˜π‘—)β€˜{𝑝}))) ∧ 𝑏 βŠ† 𝑋) ∧ 𝑝 ∈ 𝑏) β†’ ((neiβ€˜π‘—)β€˜{𝑝}) = (π‘β€˜π‘))
2928eleq2d 2817 . . . . . . . 8 (((((πœ‘ ∧ 𝑗 ∈ (TopOnβ€˜π‘‹)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((neiβ€˜π‘—)β€˜{𝑝}))) ∧ 𝑏 βŠ† 𝑋) ∧ 𝑝 ∈ 𝑏) β†’ (𝑏 ∈ ((neiβ€˜π‘—)β€˜{𝑝}) ↔ 𝑏 ∈ (π‘β€˜π‘)))
3020, 29ralbida 3265 . . . . . . 7 ((((πœ‘ ∧ 𝑗 ∈ (TopOnβ€˜π‘‹)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((neiβ€˜π‘—)β€˜{𝑝}))) ∧ 𝑏 βŠ† 𝑋) β†’ (βˆ€π‘ ∈ 𝑏 𝑏 ∈ ((neiβ€˜π‘—)β€˜{𝑝}) ↔ βˆ€π‘ ∈ 𝑏 𝑏 ∈ (π‘β€˜π‘)))
3130pm5.32da 577 . . . . . 6 (((πœ‘ ∧ 𝑗 ∈ (TopOnβ€˜π‘‹)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((neiβ€˜π‘—)β€˜{𝑝}))) β†’ ((𝑏 βŠ† 𝑋 ∧ βˆ€π‘ ∈ 𝑏 𝑏 ∈ ((neiβ€˜π‘—)β€˜{𝑝})) ↔ (𝑏 βŠ† 𝑋 ∧ βˆ€π‘ ∈ 𝑏 𝑏 ∈ (π‘β€˜π‘))))
32 toponss 22649 . . . . . . . . 9 ((𝑗 ∈ (TopOnβ€˜π‘‹) ∧ 𝑏 ∈ 𝑗) β†’ 𝑏 βŠ† 𝑋)
3332ad4ant24 750 . . . . . . . 8 ((((πœ‘ ∧ 𝑗 ∈ (TopOnβ€˜π‘‹)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((neiβ€˜π‘—)β€˜{𝑝}))) ∧ 𝑏 ∈ 𝑗) β†’ 𝑏 βŠ† 𝑋)
34 topontop 22635 . . . . . . . . . . 11 (𝑗 ∈ (TopOnβ€˜π‘‹) β†’ 𝑗 ∈ Top)
3534ad2antlr 723 . . . . . . . . . 10 (((πœ‘ ∧ 𝑗 ∈ (TopOnβ€˜π‘‹)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((neiβ€˜π‘—)β€˜{𝑝}))) β†’ 𝑗 ∈ Top)
36 opnnei 22844 . . . . . . . . . 10 (𝑗 ∈ Top β†’ (𝑏 ∈ 𝑗 ↔ βˆ€π‘ ∈ 𝑏 𝑏 ∈ ((neiβ€˜π‘—)β€˜{𝑝})))
3735, 36syl 17 . . . . . . . . 9 (((πœ‘ ∧ 𝑗 ∈ (TopOnβ€˜π‘‹)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((neiβ€˜π‘—)β€˜{𝑝}))) β†’ (𝑏 ∈ 𝑗 ↔ βˆ€π‘ ∈ 𝑏 𝑏 ∈ ((neiβ€˜π‘—)β€˜{𝑝})))
3837biimpa 475 . . . . . . . 8 ((((πœ‘ ∧ 𝑗 ∈ (TopOnβ€˜π‘‹)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((neiβ€˜π‘—)β€˜{𝑝}))) ∧ 𝑏 ∈ 𝑗) β†’ βˆ€π‘ ∈ 𝑏 𝑏 ∈ ((neiβ€˜π‘—)β€˜{𝑝}))
3933, 38jca 510 . . . . . . 7 ((((πœ‘ ∧ 𝑗 ∈ (TopOnβ€˜π‘‹)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((neiβ€˜π‘—)β€˜{𝑝}))) ∧ 𝑏 ∈ 𝑗) β†’ (𝑏 βŠ† 𝑋 ∧ βˆ€π‘ ∈ 𝑏 𝑏 ∈ ((neiβ€˜π‘—)β€˜{𝑝})))
4037biimpar 476 . . . . . . . 8 ((((πœ‘ ∧ 𝑗 ∈ (TopOnβ€˜π‘‹)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((neiβ€˜π‘—)β€˜{𝑝}))) ∧ βˆ€π‘ ∈ 𝑏 𝑏 ∈ ((neiβ€˜π‘—)β€˜{𝑝})) β†’ 𝑏 ∈ 𝑗)
4140adantrl 712 . . . . . . 7 ((((πœ‘ ∧ 𝑗 ∈ (TopOnβ€˜π‘‹)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((neiβ€˜π‘—)β€˜{𝑝}))) ∧ (𝑏 βŠ† 𝑋 ∧ βˆ€π‘ ∈ 𝑏 𝑏 ∈ ((neiβ€˜π‘—)β€˜{𝑝}))) β†’ 𝑏 ∈ 𝑗)
4239, 41impbida 797 . . . . . 6 (((πœ‘ ∧ 𝑗 ∈ (TopOnβ€˜π‘‹)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((neiβ€˜π‘—)β€˜{𝑝}))) β†’ (𝑏 ∈ 𝑗 ↔ (𝑏 βŠ† 𝑋 ∧ βˆ€π‘ ∈ 𝑏 𝑏 ∈ ((neiβ€˜π‘—)β€˜{𝑝}))))
431neipeltop 22853 . . . . . . 7 (𝑏 ∈ 𝐽 ↔ (𝑏 βŠ† 𝑋 ∧ βˆ€π‘ ∈ 𝑏 𝑏 ∈ (π‘β€˜π‘)))
4443a1i 11 . . . . . 6 (((πœ‘ ∧ 𝑗 ∈ (TopOnβ€˜π‘‹)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((neiβ€˜π‘—)β€˜{𝑝}))) β†’ (𝑏 ∈ 𝐽 ↔ (𝑏 βŠ† 𝑋 ∧ βˆ€π‘ ∈ 𝑏 𝑏 ∈ (π‘β€˜π‘))))
4531, 42, 443bitr4d 310 . . . . 5 (((πœ‘ ∧ 𝑗 ∈ (TopOnβ€˜π‘‹)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((neiβ€˜π‘—)β€˜{𝑝}))) β†’ (𝑏 ∈ 𝑗 ↔ 𝑏 ∈ 𝐽))
4645eqrdv 2728 . . . 4 (((πœ‘ ∧ 𝑗 ∈ (TopOnβ€˜π‘‹)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((neiβ€˜π‘—)β€˜{𝑝}))) β†’ 𝑗 = 𝐽)
4746ex 411 . . 3 ((πœ‘ ∧ 𝑗 ∈ (TopOnβ€˜π‘‹)) β†’ (𝑁 = (𝑝 ∈ 𝑋 ↦ ((neiβ€˜π‘—)β€˜{𝑝})) β†’ 𝑗 = 𝐽))
4847ralrimiva 3144 . 2 (πœ‘ β†’ βˆ€π‘— ∈ (TopOnβ€˜π‘‹)(𝑁 = (𝑝 ∈ 𝑋 ↦ ((neiβ€˜π‘—)β€˜{𝑝})) β†’ 𝑗 = 𝐽))
49 simpl 481 . . . . . . 7 ((𝑗 = 𝐽 ∧ 𝑝 ∈ 𝑋) β†’ 𝑗 = 𝐽)
5049fveq2d 6894 . . . . . 6 ((𝑗 = 𝐽 ∧ 𝑝 ∈ 𝑋) β†’ (neiβ€˜π‘—) = (neiβ€˜π½))
5150fveq1d 6892 . . . . 5 ((𝑗 = 𝐽 ∧ 𝑝 ∈ 𝑋) β†’ ((neiβ€˜π‘—)β€˜{𝑝}) = ((neiβ€˜π½)β€˜{𝑝}))
5251mpteq2dva 5247 . . . 4 (𝑗 = 𝐽 β†’ (𝑝 ∈ 𝑋 ↦ ((neiβ€˜π‘—)β€˜{𝑝})) = (𝑝 ∈ 𝑋 ↦ ((neiβ€˜π½)β€˜{𝑝})))
5352eqeq2d 2741 . . 3 (𝑗 = 𝐽 β†’ (𝑁 = (𝑝 ∈ 𝑋 ↦ ((neiβ€˜π‘—)β€˜{𝑝})) ↔ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((neiβ€˜π½)β€˜{𝑝}))))
5453eqreu 3724 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((neiβ€˜π½)β€˜{𝑝})) ∧ βˆ€π‘— ∈ (TopOnβ€˜π‘‹)(𝑁 = (𝑝 ∈ 𝑋 ↦ ((neiβ€˜π‘—)β€˜{𝑝})) β†’ 𝑗 = 𝐽)) β†’ βˆƒ!𝑗 ∈ (TopOnβ€˜π‘‹)𝑁 = (𝑝 ∈ 𝑋 ↦ ((neiβ€˜π‘—)β€˜{𝑝})))
5513, 14, 48, 54syl3anc 1369 1 (πœ‘ β†’ βˆƒ!𝑗 ∈ (TopOnβ€˜π‘‹)𝑁 = (𝑝 ∈ 𝑋 ↦ ((neiβ€˜π‘—)β€˜{𝑝})))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059  βˆƒwrex 3068  βˆƒ!wreu 3372  {crab 3430  Vcvv 3472   βŠ† wss 3947  π’« cpw 4601  {csn 4627  βˆͺ cuni 4907   ↦ cmpt 5230  βŸΆwf 6538  β€˜cfv 6542  ficfi 9407  Topctop 22615  TopOnctopon 22632  neicnei 22821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-om 7858  df-1o 8468  df-en 8942  df-fin 8945  df-fi 9408  df-top 22616  df-topon 22633  df-ntr 22744  df-nei 22822
This theorem is referenced by:  ustuqtop  23971
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