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Mirrors  >  Home  >  ILE Home  >  Th. List  >  neipsm GIF version
Theorem neipsm 13657
Description: A neighborhood of a set is a neighborhood of every point in the set. Proposition 1 of [BourbakiTop1] p. I.2. (Contributed by FL, 16-Nov-2006.) (Revised by Jim Kingdon, 22-Mar-2023.)
Hypothesis
Ref Expression
neips.1 𝑋 = βˆͺ 𝐽
Assertion
Ref Expression
neipsm ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ βˆƒπ‘₯ π‘₯ ∈ 𝑆) β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ↔ βˆ€π‘ ∈ 𝑆 𝑁 ∈ ((neiβ€˜π½)β€˜{𝑝})))
Distinct variable groups:   𝐽,𝑝   𝑁,𝑝   𝑆,𝑝   𝑋,𝑝   π‘₯,𝑝,𝑆
Allowed substitution hints:   𝐽(π‘₯)   𝑁(π‘₯)   𝑋(π‘₯)
Proof of Theorem neipsm
Dummy variables 𝑔 β„Ž 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snssi 3737 . . . . . 6 (𝑝 ∈ 𝑆 β†’ {𝑝} βŠ† 𝑆)
2 neiss 13653 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ∧ {𝑝} βŠ† 𝑆) β†’ 𝑁 ∈ ((neiβ€˜π½)β€˜{𝑝}))
31, 2syl3an3 1273 . . . . 5 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ∧ 𝑝 ∈ 𝑆) β†’ 𝑁 ∈ ((neiβ€˜π½)β€˜{𝑝}))
433exp 1202 . . . 4 (𝐽 ∈ Top β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) β†’ (𝑝 ∈ 𝑆 β†’ 𝑁 ∈ ((neiβ€˜π½)β€˜{𝑝}))))
54ralrimdv 2556 . . 3 (𝐽 ∈ Top β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) β†’ βˆ€π‘ ∈ 𝑆 𝑁 ∈ ((neiβ€˜π½)β€˜{𝑝})))
653ad2ant1 1018 . 2 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ βˆƒπ‘₯ π‘₯ ∈ 𝑆) β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) β†’ βˆ€π‘ ∈ 𝑆 𝑁 ∈ ((neiβ€˜π½)β€˜{𝑝})))
7 eleq1w 2238 . . . . . . 7 (𝑝 = π‘₯ β†’ (𝑝 ∈ 𝑆 ↔ π‘₯ ∈ 𝑆))
87cbvexv 1918 . . . . . 6 (βˆƒπ‘ 𝑝 ∈ 𝑆 ↔ βˆƒπ‘₯ π‘₯ ∈ 𝑆)
9 r19.28mv 3516 . . . . . 6 (βˆƒπ‘ 𝑝 ∈ 𝑆 β†’ (βˆ€π‘ ∈ 𝑆 (𝑁 βŠ† 𝑋 ∧ βˆƒπ‘” ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁)) ↔ (𝑁 βŠ† 𝑋 ∧ βˆ€π‘ ∈ 𝑆 βˆƒπ‘” ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁))))
108, 9sylbir 135 . . . . 5 (βˆƒπ‘₯ π‘₯ ∈ 𝑆 β†’ (βˆ€π‘ ∈ 𝑆 (𝑁 βŠ† 𝑋 ∧ βˆƒπ‘” ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁)) ↔ (𝑁 βŠ† 𝑋 ∧ βˆ€π‘ ∈ 𝑆 βˆƒπ‘” ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁))))
11103ad2ant3 1020 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ βˆƒπ‘₯ π‘₯ ∈ 𝑆) β†’ (βˆ€π‘ ∈ 𝑆 (𝑁 βŠ† 𝑋 ∧ βˆƒπ‘” ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁)) ↔ (𝑁 βŠ† 𝑋 ∧ βˆ€π‘ ∈ 𝑆 βˆƒπ‘” ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁))))
12 ssrab2 3241 . . . . . . . . . 10 {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} βŠ† 𝐽
13 uniopn 13504 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} βŠ† 𝐽) β†’ βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} ∈ 𝐽)
1412, 13mpan2 425 . . . . . . . . 9 (𝐽 ∈ Top β†’ βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} ∈ 𝐽)
1514ad2antrr 488 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ βˆ€π‘ ∈ 𝑆 βˆƒπ‘” ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁)) β†’ βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} ∈ 𝐽)
16 sseq1 3179 . . . . . . . . . . . . . . . 16 (𝑣 = 𝑔 β†’ (𝑣 βŠ† 𝑁 ↔ 𝑔 βŠ† 𝑁))
1716elrab 2894 . . . . . . . . . . . . . . 15 (𝑔 ∈ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} ↔ (𝑔 ∈ 𝐽 ∧ 𝑔 βŠ† 𝑁))
18 elunii 3815 . . . . . . . . . . . . . . 15 ((𝑝 ∈ 𝑔 ∧ 𝑔 ∈ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁}) β†’ 𝑝 ∈ βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁})
1917, 18sylan2br 288 . . . . . . . . . . . . . 14 ((𝑝 ∈ 𝑔 ∧ (𝑔 ∈ 𝐽 ∧ 𝑔 βŠ† 𝑁)) β†’ 𝑝 ∈ βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁})
2019an12s 565 . . . . . . . . . . . . 13 ((𝑔 ∈ 𝐽 ∧ (𝑝 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁)) β†’ 𝑝 ∈ βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁})
2120rexlimiva 2589 . . . . . . . . . . . 12 (βˆƒπ‘” ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁) β†’ 𝑝 ∈ βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁})
2221ralimi 2540 . . . . . . . . . . 11 (βˆ€π‘ ∈ 𝑆 βˆƒπ‘” ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁) β†’ βˆ€π‘ ∈ 𝑆 𝑝 ∈ βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁})
23 dfss3 3146 . . . . . . . . . . 11 (𝑆 βŠ† βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} ↔ βˆ€π‘ ∈ 𝑆 𝑝 ∈ βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁})
2422, 23sylibr 134 . . . . . . . . . 10 (βˆ€π‘ ∈ 𝑆 βˆƒπ‘” ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁) β†’ 𝑆 βŠ† βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁})
2524adantl 277 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ βˆ€π‘ ∈ 𝑆 βˆƒπ‘” ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁)) β†’ 𝑆 βŠ† βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁})
26 unissb 3840 . . . . . . . . . 10 (βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} βŠ† 𝑁 ↔ βˆ€β„Ž ∈ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁}β„Ž βŠ† 𝑁)
27 sseq1 3179 . . . . . . . . . . . 12 (𝑣 = β„Ž β†’ (𝑣 βŠ† 𝑁 ↔ β„Ž βŠ† 𝑁))
2827elrab 2894 . . . . . . . . . . 11 (β„Ž ∈ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} ↔ (β„Ž ∈ 𝐽 ∧ β„Ž βŠ† 𝑁))
2928simprbi 275 . . . . . . . . . 10 (β„Ž ∈ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} β†’ β„Ž βŠ† 𝑁)
3026, 29mprgbir 2535 . . . . . . . . 9 βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} βŠ† 𝑁
3125, 30jctir 313 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ βˆ€π‘ ∈ 𝑆 βˆƒπ‘” ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁)) β†’ (𝑆 βŠ† βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} ∧ βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} βŠ† 𝑁))
32 sseq2 3180 . . . . . . . . . 10 (β„Ž = βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} β†’ (𝑆 βŠ† β„Ž ↔ 𝑆 βŠ† βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁}))
33 sseq1 3179 . . . . . . . . . 10 (β„Ž = βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} β†’ (β„Ž βŠ† 𝑁 ↔ βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} βŠ† 𝑁))
3432, 33anbi12d 473 . . . . . . . . 9 (β„Ž = βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} β†’ ((𝑆 βŠ† β„Ž ∧ β„Ž βŠ† 𝑁) ↔ (𝑆 βŠ† βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} ∧ βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} βŠ† 𝑁)))
3534rspcev 2842 . . . . . . . 8 ((βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} ∈ 𝐽 ∧ (𝑆 βŠ† βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} ∧ βˆͺ {𝑣 ∈ 𝐽 ∣ 𝑣 βŠ† 𝑁} βŠ† 𝑁)) β†’ βˆƒβ„Ž ∈ 𝐽 (𝑆 βŠ† β„Ž ∧ β„Ž βŠ† 𝑁))
3615, 31, 35syl2anc 411 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ βˆ€π‘ ∈ 𝑆 βˆƒπ‘” ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁)) β†’ βˆƒβ„Ž ∈ 𝐽 (𝑆 βŠ† β„Ž ∧ β„Ž βŠ† 𝑁))
3736ex 115 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ (βˆ€π‘ ∈ 𝑆 βˆƒπ‘” ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁) β†’ βˆƒβ„Ž ∈ 𝐽 (𝑆 βŠ† β„Ž ∧ β„Ž βŠ† 𝑁)))
3837anim2d 337 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((𝑁 βŠ† 𝑋 ∧ βˆ€π‘ ∈ 𝑆 βˆƒπ‘” ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁)) β†’ (𝑁 βŠ† 𝑋 ∧ βˆƒβ„Ž ∈ 𝐽 (𝑆 βŠ† β„Ž ∧ β„Ž βŠ† 𝑁))))
39383adant3 1017 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ βˆƒπ‘₯ π‘₯ ∈ 𝑆) β†’ ((𝑁 βŠ† 𝑋 ∧ βˆ€π‘ ∈ 𝑆 βˆƒπ‘” ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁)) β†’ (𝑁 βŠ† 𝑋 ∧ βˆƒβ„Ž ∈ 𝐽 (𝑆 βŠ† β„Ž ∧ β„Ž βŠ† 𝑁))))
4011, 39sylbid 150 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ βˆƒπ‘₯ π‘₯ ∈ 𝑆) β†’ (βˆ€π‘ ∈ 𝑆 (𝑁 βŠ† 𝑋 ∧ βˆƒπ‘” ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁)) β†’ (𝑁 βŠ† 𝑋 ∧ βˆƒβ„Ž ∈ 𝐽 (𝑆 βŠ† β„Ž ∧ β„Ž βŠ† 𝑁))))
41 ssel2 3151 . . . . . . 7 ((𝑆 βŠ† 𝑋 ∧ 𝑝 ∈ 𝑆) β†’ 𝑝 ∈ 𝑋)
42 neips.1 . . . . . . . 8 𝑋 = βˆͺ 𝐽
4342isneip 13649 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑝 ∈ 𝑋) β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜{𝑝}) ↔ (𝑁 βŠ† 𝑋 ∧ βˆƒπ‘” ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁))))
4441, 43sylan2 286 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑆 βŠ† 𝑋 ∧ 𝑝 ∈ 𝑆)) β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜{𝑝}) ↔ (𝑁 βŠ† 𝑋 ∧ βˆƒπ‘” ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁))))
4544anassrs 400 . . . . 5 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ 𝑝 ∈ 𝑆) β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜{𝑝}) ↔ (𝑁 βŠ† 𝑋 ∧ βˆƒπ‘” ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁))))
4645ralbidva 2473 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ (βˆ€π‘ ∈ 𝑆 𝑁 ∈ ((neiβ€˜π½)β€˜{𝑝}) ↔ βˆ€π‘ ∈ 𝑆 (𝑁 βŠ† 𝑋 ∧ βˆƒπ‘” ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁))))
47463adant3 1017 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ βˆƒπ‘₯ π‘₯ ∈ 𝑆) β†’ (βˆ€π‘ ∈ 𝑆 𝑁 ∈ ((neiβ€˜π½)β€˜{𝑝}) ↔ βˆ€π‘ ∈ 𝑆 (𝑁 βŠ† 𝑋 ∧ βˆƒπ‘” ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 βŠ† 𝑁))))
4842isnei 13647 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ↔ (𝑁 βŠ† 𝑋 ∧ βˆƒβ„Ž ∈ 𝐽 (𝑆 βŠ† β„Ž ∧ β„Ž βŠ† 𝑁))))
49483adant3 1017 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ βˆƒπ‘₯ π‘₯ ∈ 𝑆) β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ↔ (𝑁 βŠ† 𝑋 ∧ βˆƒβ„Ž ∈ 𝐽 (𝑆 βŠ† β„Ž ∧ β„Ž βŠ† 𝑁))))
5040, 47, 493imtr4d 203 . 2 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ βˆƒπ‘₯ π‘₯ ∈ 𝑆) β†’ (βˆ€π‘ ∈ 𝑆 𝑁 ∈ ((neiβ€˜π½)β€˜{𝑝}) β†’ 𝑁 ∈ ((neiβ€˜π½)β€˜π‘†)))
516, 50impbid 129 1 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ βˆƒπ‘₯ π‘₯ ∈ 𝑆) β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) ↔ βˆ€π‘ ∈ 𝑆 𝑁 ∈ ((neiβ€˜π½)β€˜{𝑝})))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 978   = wceq 1353  βˆƒwex 1492   ∈ wcel 2148  βˆ€wral 2455  βˆƒwrex 2456  {crab 2459   βŠ† wss 3130  {csn 3593  βˆͺ cuni 3810  β€˜cfv 5217  Topctop 13500  neicnei 13641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-pow 4175  ax-pr 4210
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-top 13501  df-nei 13642
This theorem is referenced by: (None)
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