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Theorem nllyidm 22863
Description: Idempotence of the "n-locally" predicate, i.e. being "n-locally 𝐴 " is a local property. (Use loclly 22861 to show 𝑛-Locally 𝑛-Locally 𝐴 = 𝑛-Locally 𝐴.) (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
nllyidm Locally 𝑛-Locally 𝐴 = 𝑛-Locally 𝐴

Proof of Theorem nllyidm
Dummy variables 𝑗 𝑒 𝑣 π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 llytop 22846 . . . 4 (𝑗 ∈ Locally 𝑛-Locally 𝐴 β†’ 𝑗 ∈ Top)
2 llyi 22848 . . . . . . 7 ((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ π‘₯ ∈ 𝑗 ∧ 𝑦 ∈ π‘₯) β†’ βˆƒπ‘’ ∈ 𝑗 (𝑒 βŠ† π‘₯ ∧ 𝑦 ∈ 𝑒 ∧ (𝑗 β†Ύt 𝑒) ∈ 𝑛-Locally 𝐴))
3 simprr3 1224 . . . . . . . . 9 (((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ π‘₯ ∈ 𝑗 ∧ 𝑦 ∈ π‘₯) ∧ (𝑒 ∈ 𝑗 ∧ (𝑒 βŠ† π‘₯ ∧ 𝑦 ∈ 𝑒 ∧ (𝑗 β†Ύt 𝑒) ∈ 𝑛-Locally 𝐴))) β†’ (𝑗 β†Ύt 𝑒) ∈ 𝑛-Locally 𝐴)
4 simprl 770 . . . . . . . . . 10 (((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ π‘₯ ∈ 𝑗 ∧ 𝑦 ∈ π‘₯) ∧ (𝑒 ∈ 𝑗 ∧ (𝑒 βŠ† π‘₯ ∧ 𝑦 ∈ 𝑒 ∧ (𝑗 β†Ύt 𝑒) ∈ 𝑛-Locally 𝐴))) β†’ 𝑒 ∈ 𝑗)
5 ssidd 3971 . . . . . . . . . 10 (((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ π‘₯ ∈ 𝑗 ∧ 𝑦 ∈ π‘₯) ∧ (𝑒 ∈ 𝑗 ∧ (𝑒 βŠ† π‘₯ ∧ 𝑦 ∈ 𝑒 ∧ (𝑗 β†Ύt 𝑒) ∈ 𝑛-Locally 𝐴))) β†’ 𝑒 βŠ† 𝑒)
6 simpl1 1192 . . . . . . . . . . . 12 (((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ π‘₯ ∈ 𝑗 ∧ 𝑦 ∈ π‘₯) ∧ (𝑒 ∈ 𝑗 ∧ (𝑒 βŠ† π‘₯ ∧ 𝑦 ∈ 𝑒 ∧ (𝑗 β†Ύt 𝑒) ∈ 𝑛-Locally 𝐴))) β†’ 𝑗 ∈ Locally 𝑛-Locally 𝐴)
76, 1syl 17 . . . . . . . . . . 11 (((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ π‘₯ ∈ 𝑗 ∧ 𝑦 ∈ π‘₯) ∧ (𝑒 ∈ 𝑗 ∧ (𝑒 βŠ† π‘₯ ∧ 𝑦 ∈ 𝑒 ∧ (𝑗 β†Ύt 𝑒) ∈ 𝑛-Locally 𝐴))) β†’ 𝑗 ∈ Top)
8 restopn2 22551 . . . . . . . . . . 11 ((𝑗 ∈ Top ∧ 𝑒 ∈ 𝑗) β†’ (𝑒 ∈ (𝑗 β†Ύt 𝑒) ↔ (𝑒 ∈ 𝑗 ∧ 𝑒 βŠ† 𝑒)))
97, 4, 8syl2anc 585 . . . . . . . . . 10 (((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ π‘₯ ∈ 𝑗 ∧ 𝑦 ∈ π‘₯) ∧ (𝑒 ∈ 𝑗 ∧ (𝑒 βŠ† π‘₯ ∧ 𝑦 ∈ 𝑒 ∧ (𝑗 β†Ύt 𝑒) ∈ 𝑛-Locally 𝐴))) β†’ (𝑒 ∈ (𝑗 β†Ύt 𝑒) ↔ (𝑒 ∈ 𝑗 ∧ 𝑒 βŠ† 𝑒)))
104, 5, 9mpbir2and 712 . . . . . . . . 9 (((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ π‘₯ ∈ 𝑗 ∧ 𝑦 ∈ π‘₯) ∧ (𝑒 ∈ 𝑗 ∧ (𝑒 βŠ† π‘₯ ∧ 𝑦 ∈ 𝑒 ∧ (𝑗 β†Ύt 𝑒) ∈ 𝑛-Locally 𝐴))) β†’ 𝑒 ∈ (𝑗 β†Ύt 𝑒))
11 simprr2 1223 . . . . . . . . 9 (((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ π‘₯ ∈ 𝑗 ∧ 𝑦 ∈ π‘₯) ∧ (𝑒 ∈ 𝑗 ∧ (𝑒 βŠ† π‘₯ ∧ 𝑦 ∈ 𝑒 ∧ (𝑗 β†Ύt 𝑒) ∈ 𝑛-Locally 𝐴))) β†’ 𝑦 ∈ 𝑒)
12 nlly2i 22850 . . . . . . . . 9 (((𝑗 β†Ύt 𝑒) ∈ 𝑛-Locally 𝐴 ∧ 𝑒 ∈ (𝑗 β†Ύt 𝑒) ∧ 𝑦 ∈ 𝑒) β†’ βˆƒπ‘£ ∈ 𝒫 π‘’βˆƒπ‘§ ∈ (𝑗 β†Ύt 𝑒)(𝑦 ∈ 𝑧 ∧ 𝑧 βŠ† 𝑣 ∧ ((𝑗 β†Ύt 𝑒) β†Ύt 𝑣) ∈ 𝐴))
133, 10, 11, 12syl3anc 1372 . . . . . . . 8 (((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ π‘₯ ∈ 𝑗 ∧ 𝑦 ∈ π‘₯) ∧ (𝑒 ∈ 𝑗 ∧ (𝑒 βŠ† π‘₯ ∧ 𝑦 ∈ 𝑒 ∧ (𝑗 β†Ύt 𝑒) ∈ 𝑛-Locally 𝐴))) β†’ βˆƒπ‘£ ∈ 𝒫 π‘’βˆƒπ‘§ ∈ (𝑗 β†Ύt 𝑒)(𝑦 ∈ 𝑧 ∧ 𝑧 βŠ† 𝑣 ∧ ((𝑗 β†Ύt 𝑒) β†Ύt 𝑣) ∈ 𝐴))
14 restopn2 22551 . . . . . . . . . . . . . 14 ((𝑗 ∈ Top ∧ 𝑒 ∈ 𝑗) β†’ (𝑧 ∈ (𝑗 β†Ύt 𝑒) ↔ (𝑧 ∈ 𝑗 ∧ 𝑧 βŠ† 𝑒)))
157, 4, 14syl2anc 585 . . . . . . . . . . . . 13 (((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ π‘₯ ∈ 𝑗 ∧ 𝑦 ∈ π‘₯) ∧ (𝑒 ∈ 𝑗 ∧ (𝑒 βŠ† π‘₯ ∧ 𝑦 ∈ 𝑒 ∧ (𝑗 β†Ύt 𝑒) ∈ 𝑛-Locally 𝐴))) β†’ (𝑧 ∈ (𝑗 β†Ύt 𝑒) ↔ (𝑧 ∈ 𝑗 ∧ 𝑧 βŠ† 𝑒)))
1615adantr 482 . . . . . . . . . . . 12 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ π‘₯ ∈ 𝑗 ∧ 𝑦 ∈ π‘₯) ∧ (𝑒 ∈ 𝑗 ∧ (𝑒 βŠ† π‘₯ ∧ 𝑦 ∈ 𝑒 ∧ (𝑗 β†Ύt 𝑒) ∈ 𝑛-Locally 𝐴))) ∧ 𝑣 ∈ 𝒫 𝑒) β†’ (𝑧 ∈ (𝑗 β†Ύt 𝑒) ↔ (𝑧 ∈ 𝑗 ∧ 𝑧 βŠ† 𝑒)))
177adantr 482 . . . . . . . . . . . . . . . . 17 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ π‘₯ ∈ 𝑗 ∧ 𝑦 ∈ π‘₯) ∧ (𝑒 ∈ 𝑗 ∧ (𝑒 βŠ† π‘₯ ∧ 𝑦 ∈ 𝑒 ∧ (𝑗 β†Ύt 𝑒) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑒 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 βŠ† 𝑒) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 βŠ† 𝑣 ∧ ((𝑗 β†Ύt 𝑒) β†Ύt 𝑣) ∈ 𝐴))) β†’ 𝑗 ∈ Top)
18 simpr2l 1233 . . . . . . . . . . . . . . . . . 18 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ π‘₯ ∈ 𝑗 ∧ 𝑦 ∈ π‘₯) ∧ (𝑒 ∈ 𝑗 ∧ (𝑒 βŠ† π‘₯ ∧ 𝑦 ∈ 𝑒 ∧ (𝑗 β†Ύt 𝑒) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑒 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 βŠ† 𝑒) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 βŠ† 𝑣 ∧ ((𝑗 β†Ύt 𝑒) β†Ύt 𝑣) ∈ 𝐴))) β†’ 𝑧 ∈ 𝑗)
19 simpr31 1264 . . . . . . . . . . . . . . . . . 18 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ π‘₯ ∈ 𝑗 ∧ 𝑦 ∈ π‘₯) ∧ (𝑒 ∈ 𝑗 ∧ (𝑒 βŠ† π‘₯ ∧ 𝑦 ∈ 𝑒 ∧ (𝑗 β†Ύt 𝑒) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑒 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 βŠ† 𝑒) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 βŠ† 𝑣 ∧ ((𝑗 β†Ύt 𝑒) β†Ύt 𝑣) ∈ 𝐴))) β†’ 𝑦 ∈ 𝑧)
20 opnneip 22493 . . . . . . . . . . . . . . . . . 18 ((𝑗 ∈ Top ∧ 𝑧 ∈ 𝑗 ∧ 𝑦 ∈ 𝑧) β†’ 𝑧 ∈ ((neiβ€˜π‘—)β€˜{𝑦}))
2117, 18, 19, 20syl3anc 1372 . . . . . . . . . . . . . . . . 17 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ π‘₯ ∈ 𝑗 ∧ 𝑦 ∈ π‘₯) ∧ (𝑒 ∈ 𝑗 ∧ (𝑒 βŠ† π‘₯ ∧ 𝑦 ∈ 𝑒 ∧ (𝑗 β†Ύt 𝑒) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑒 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 βŠ† 𝑒) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 βŠ† 𝑣 ∧ ((𝑗 β†Ύt 𝑒) β†Ύt 𝑣) ∈ 𝐴))) β†’ 𝑧 ∈ ((neiβ€˜π‘—)β€˜{𝑦}))
22 simpr32 1265 . . . . . . . . . . . . . . . . 17 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ π‘₯ ∈ 𝑗 ∧ 𝑦 ∈ π‘₯) ∧ (𝑒 ∈ 𝑗 ∧ (𝑒 βŠ† π‘₯ ∧ 𝑦 ∈ 𝑒 ∧ (𝑗 β†Ύt 𝑒) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑒 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 βŠ† 𝑒) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 βŠ† 𝑣 ∧ ((𝑗 β†Ύt 𝑒) β†Ύt 𝑣) ∈ 𝐴))) β†’ 𝑧 βŠ† 𝑣)
23 simpr1 1195 . . . . . . . . . . . . . . . . . . 19 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ π‘₯ ∈ 𝑗 ∧ 𝑦 ∈ π‘₯) ∧ (𝑒 ∈ 𝑗 ∧ (𝑒 βŠ† π‘₯ ∧ 𝑦 ∈ 𝑒 ∧ (𝑗 β†Ύt 𝑒) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑒 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 βŠ† 𝑒) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 βŠ† 𝑣 ∧ ((𝑗 β†Ύt 𝑒) β†Ύt 𝑣) ∈ 𝐴))) β†’ 𝑣 ∈ 𝒫 𝑒)
2423elpwid 4573 . . . . . . . . . . . . . . . . . 18 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ π‘₯ ∈ 𝑗 ∧ 𝑦 ∈ π‘₯) ∧ (𝑒 ∈ 𝑗 ∧ (𝑒 βŠ† π‘₯ ∧ 𝑦 ∈ 𝑒 ∧ (𝑗 β†Ύt 𝑒) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑒 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 βŠ† 𝑒) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 βŠ† 𝑣 ∧ ((𝑗 β†Ύt 𝑒) β†Ύt 𝑣) ∈ 𝐴))) β†’ 𝑣 βŠ† 𝑒)
254adantr 482 . . . . . . . . . . . . . . . . . . 19 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ π‘₯ ∈ 𝑗 ∧ 𝑦 ∈ π‘₯) ∧ (𝑒 ∈ 𝑗 ∧ (𝑒 βŠ† π‘₯ ∧ 𝑦 ∈ 𝑒 ∧ (𝑗 β†Ύt 𝑒) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑒 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 βŠ† 𝑒) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 βŠ† 𝑣 ∧ ((𝑗 β†Ύt 𝑒) β†Ύt 𝑣) ∈ 𝐴))) β†’ 𝑒 ∈ 𝑗)
26 elssuni 4902 . . . . . . . . . . . . . . . . . . 19 (𝑒 ∈ 𝑗 β†’ 𝑒 βŠ† βˆͺ 𝑗)
2725, 26syl 17 . . . . . . . . . . . . . . . . . 18 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ π‘₯ ∈ 𝑗 ∧ 𝑦 ∈ π‘₯) ∧ (𝑒 ∈ 𝑗 ∧ (𝑒 βŠ† π‘₯ ∧ 𝑦 ∈ 𝑒 ∧ (𝑗 β†Ύt 𝑒) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑒 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 βŠ† 𝑒) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 βŠ† 𝑣 ∧ ((𝑗 β†Ύt 𝑒) β†Ύt 𝑣) ∈ 𝐴))) β†’ 𝑒 βŠ† βˆͺ 𝑗)
2824, 27sstrd 3958 . . . . . . . . . . . . . . . . 17 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ π‘₯ ∈ 𝑗 ∧ 𝑦 ∈ π‘₯) ∧ (𝑒 ∈ 𝑗 ∧ (𝑒 βŠ† π‘₯ ∧ 𝑦 ∈ 𝑒 ∧ (𝑗 β†Ύt 𝑒) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑒 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 βŠ† 𝑒) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 βŠ† 𝑣 ∧ ((𝑗 β†Ύt 𝑒) β†Ύt 𝑣) ∈ 𝐴))) β†’ 𝑣 βŠ† βˆͺ 𝑗)
29 eqid 2733 . . . . . . . . . . . . . . . . . 18 βˆͺ 𝑗 = βˆͺ 𝑗
3029ssnei2 22490 . . . . . . . . . . . . . . . . 17 (((𝑗 ∈ Top ∧ 𝑧 ∈ ((neiβ€˜π‘—)β€˜{𝑦})) ∧ (𝑧 βŠ† 𝑣 ∧ 𝑣 βŠ† βˆͺ 𝑗)) β†’ 𝑣 ∈ ((neiβ€˜π‘—)β€˜{𝑦}))
3117, 21, 22, 28, 30syl22anc 838 . . . . . . . . . . . . . . . 16 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ π‘₯ ∈ 𝑗 ∧ 𝑦 ∈ π‘₯) ∧ (𝑒 ∈ 𝑗 ∧ (𝑒 βŠ† π‘₯ ∧ 𝑦 ∈ 𝑒 ∧ (𝑗 β†Ύt 𝑒) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑒 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 βŠ† 𝑒) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 βŠ† 𝑣 ∧ ((𝑗 β†Ύt 𝑒) β†Ύt 𝑣) ∈ 𝐴))) β†’ 𝑣 ∈ ((neiβ€˜π‘—)β€˜{𝑦}))
32 simprr1 1222 . . . . . . . . . . . . . . . . . . 19 (((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ π‘₯ ∈ 𝑗 ∧ 𝑦 ∈ π‘₯) ∧ (𝑒 ∈ 𝑗 ∧ (𝑒 βŠ† π‘₯ ∧ 𝑦 ∈ 𝑒 ∧ (𝑗 β†Ύt 𝑒) ∈ 𝑛-Locally 𝐴))) β†’ 𝑒 βŠ† π‘₯)
3332adantr 482 . . . . . . . . . . . . . . . . . 18 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ π‘₯ ∈ 𝑗 ∧ 𝑦 ∈ π‘₯) ∧ (𝑒 ∈ 𝑗 ∧ (𝑒 βŠ† π‘₯ ∧ 𝑦 ∈ 𝑒 ∧ (𝑗 β†Ύt 𝑒) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑒 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 βŠ† 𝑒) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 βŠ† 𝑣 ∧ ((𝑗 β†Ύt 𝑒) β†Ύt 𝑣) ∈ 𝐴))) β†’ 𝑒 βŠ† π‘₯)
3424, 33sstrd 3958 . . . . . . . . . . . . . . . . 17 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ π‘₯ ∈ 𝑗 ∧ 𝑦 ∈ π‘₯) ∧ (𝑒 ∈ 𝑗 ∧ (𝑒 βŠ† π‘₯ ∧ 𝑦 ∈ 𝑒 ∧ (𝑗 β†Ύt 𝑒) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑒 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 βŠ† 𝑒) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 βŠ† 𝑣 ∧ ((𝑗 β†Ύt 𝑒) β†Ύt 𝑣) ∈ 𝐴))) β†’ 𝑣 βŠ† π‘₯)
35 velpw 4569 . . . . . . . . . . . . . . . . 17 (𝑣 ∈ 𝒫 π‘₯ ↔ 𝑣 βŠ† π‘₯)
3634, 35sylibr 233 . . . . . . . . . . . . . . . 16 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ π‘₯ ∈ 𝑗 ∧ 𝑦 ∈ π‘₯) ∧ (𝑒 ∈ 𝑗 ∧ (𝑒 βŠ† π‘₯ ∧ 𝑦 ∈ 𝑒 ∧ (𝑗 β†Ύt 𝑒) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑒 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 βŠ† 𝑒) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 βŠ† 𝑣 ∧ ((𝑗 β†Ύt 𝑒) β†Ύt 𝑣) ∈ 𝐴))) β†’ 𝑣 ∈ 𝒫 π‘₯)
3731, 36elind 4158 . . . . . . . . . . . . . . 15 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ π‘₯ ∈ 𝑗 ∧ 𝑦 ∈ π‘₯) ∧ (𝑒 ∈ 𝑗 ∧ (𝑒 βŠ† π‘₯ ∧ 𝑦 ∈ 𝑒 ∧ (𝑗 β†Ύt 𝑒) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑒 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 βŠ† 𝑒) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 βŠ† 𝑣 ∧ ((𝑗 β†Ύt 𝑒) β†Ύt 𝑣) ∈ 𝐴))) β†’ 𝑣 ∈ (((neiβ€˜π‘—)β€˜{𝑦}) ∩ 𝒫 π‘₯))
38 restabs 22539 . . . . . . . . . . . . . . . . 17 ((𝑗 ∈ Top ∧ 𝑣 βŠ† 𝑒 ∧ 𝑒 ∈ 𝑗) β†’ ((𝑗 β†Ύt 𝑒) β†Ύt 𝑣) = (𝑗 β†Ύt 𝑣))
3917, 24, 25, 38syl3anc 1372 . . . . . . . . . . . . . . . 16 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ π‘₯ ∈ 𝑗 ∧ 𝑦 ∈ π‘₯) ∧ (𝑒 ∈ 𝑗 ∧ (𝑒 βŠ† π‘₯ ∧ 𝑦 ∈ 𝑒 ∧ (𝑗 β†Ύt 𝑒) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑒 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 βŠ† 𝑒) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 βŠ† 𝑣 ∧ ((𝑗 β†Ύt 𝑒) β†Ύt 𝑣) ∈ 𝐴))) β†’ ((𝑗 β†Ύt 𝑒) β†Ύt 𝑣) = (𝑗 β†Ύt 𝑣))
40 simpr33 1266 . . . . . . . . . . . . . . . 16 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ π‘₯ ∈ 𝑗 ∧ 𝑦 ∈ π‘₯) ∧ (𝑒 ∈ 𝑗 ∧ (𝑒 βŠ† π‘₯ ∧ 𝑦 ∈ 𝑒 ∧ (𝑗 β†Ύt 𝑒) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑒 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 βŠ† 𝑒) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 βŠ† 𝑣 ∧ ((𝑗 β†Ύt 𝑒) β†Ύt 𝑣) ∈ 𝐴))) β†’ ((𝑗 β†Ύt 𝑒) β†Ύt 𝑣) ∈ 𝐴)
4139, 40eqeltrrd 2835 . . . . . . . . . . . . . . 15 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ π‘₯ ∈ 𝑗 ∧ 𝑦 ∈ π‘₯) ∧ (𝑒 ∈ 𝑗 ∧ (𝑒 βŠ† π‘₯ ∧ 𝑦 ∈ 𝑒 ∧ (𝑗 β†Ύt 𝑒) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑒 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 βŠ† 𝑒) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 βŠ† 𝑣 ∧ ((𝑗 β†Ύt 𝑒) β†Ύt 𝑣) ∈ 𝐴))) β†’ (𝑗 β†Ύt 𝑣) ∈ 𝐴)
4237, 41jca 513 . . . . . . . . . . . . . 14 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ π‘₯ ∈ 𝑗 ∧ 𝑦 ∈ π‘₯) ∧ (𝑒 ∈ 𝑗 ∧ (𝑒 βŠ† π‘₯ ∧ 𝑦 ∈ 𝑒 ∧ (𝑗 β†Ύt 𝑒) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑒 ∧ (𝑧 ∈ 𝑗 ∧ 𝑧 βŠ† 𝑒) ∧ (𝑦 ∈ 𝑧 ∧ 𝑧 βŠ† 𝑣 ∧ ((𝑗 β†Ύt 𝑒) β†Ύt 𝑣) ∈ 𝐴))) β†’ (𝑣 ∈ (((neiβ€˜π‘—)β€˜{𝑦}) ∩ 𝒫 π‘₯) ∧ (𝑗 β†Ύt 𝑣) ∈ 𝐴))
43423exp2 1355 . . . . . . . . . . . . 13 (((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ π‘₯ ∈ 𝑗 ∧ 𝑦 ∈ π‘₯) ∧ (𝑒 ∈ 𝑗 ∧ (𝑒 βŠ† π‘₯ ∧ 𝑦 ∈ 𝑒 ∧ (𝑗 β†Ύt 𝑒) ∈ 𝑛-Locally 𝐴))) β†’ (𝑣 ∈ 𝒫 𝑒 β†’ ((𝑧 ∈ 𝑗 ∧ 𝑧 βŠ† 𝑒) β†’ ((𝑦 ∈ 𝑧 ∧ 𝑧 βŠ† 𝑣 ∧ ((𝑗 β†Ύt 𝑒) β†Ύt 𝑣) ∈ 𝐴) β†’ (𝑣 ∈ (((neiβ€˜π‘—)β€˜{𝑦}) ∩ 𝒫 π‘₯) ∧ (𝑗 β†Ύt 𝑣) ∈ 𝐴)))))
4443imp 408 . . . . . . . . . . . 12 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ π‘₯ ∈ 𝑗 ∧ 𝑦 ∈ π‘₯) ∧ (𝑒 ∈ 𝑗 ∧ (𝑒 βŠ† π‘₯ ∧ 𝑦 ∈ 𝑒 ∧ (𝑗 β†Ύt 𝑒) ∈ 𝑛-Locally 𝐴))) ∧ 𝑣 ∈ 𝒫 𝑒) β†’ ((𝑧 ∈ 𝑗 ∧ 𝑧 βŠ† 𝑒) β†’ ((𝑦 ∈ 𝑧 ∧ 𝑧 βŠ† 𝑣 ∧ ((𝑗 β†Ύt 𝑒) β†Ύt 𝑣) ∈ 𝐴) β†’ (𝑣 ∈ (((neiβ€˜π‘—)β€˜{𝑦}) ∩ 𝒫 π‘₯) ∧ (𝑗 β†Ύt 𝑣) ∈ 𝐴))))
4516, 44sylbid 239 . . . . . . . . . . 11 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ π‘₯ ∈ 𝑗 ∧ 𝑦 ∈ π‘₯) ∧ (𝑒 ∈ 𝑗 ∧ (𝑒 βŠ† π‘₯ ∧ 𝑦 ∈ 𝑒 ∧ (𝑗 β†Ύt 𝑒) ∈ 𝑛-Locally 𝐴))) ∧ 𝑣 ∈ 𝒫 𝑒) β†’ (𝑧 ∈ (𝑗 β†Ύt 𝑒) β†’ ((𝑦 ∈ 𝑧 ∧ 𝑧 βŠ† 𝑣 ∧ ((𝑗 β†Ύt 𝑒) β†Ύt 𝑣) ∈ 𝐴) β†’ (𝑣 ∈ (((neiβ€˜π‘—)β€˜{𝑦}) ∩ 𝒫 π‘₯) ∧ (𝑗 β†Ύt 𝑣) ∈ 𝐴))))
4645rexlimdv 3147 . . . . . . . . . 10 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ π‘₯ ∈ 𝑗 ∧ 𝑦 ∈ π‘₯) ∧ (𝑒 ∈ 𝑗 ∧ (𝑒 βŠ† π‘₯ ∧ 𝑦 ∈ 𝑒 ∧ (𝑗 β†Ύt 𝑒) ∈ 𝑛-Locally 𝐴))) ∧ 𝑣 ∈ 𝒫 𝑒) β†’ (βˆƒπ‘§ ∈ (𝑗 β†Ύt 𝑒)(𝑦 ∈ 𝑧 ∧ 𝑧 βŠ† 𝑣 ∧ ((𝑗 β†Ύt 𝑒) β†Ύt 𝑣) ∈ 𝐴) β†’ (𝑣 ∈ (((neiβ€˜π‘—)β€˜{𝑦}) ∩ 𝒫 π‘₯) ∧ (𝑗 β†Ύt 𝑣) ∈ 𝐴)))
4746expimpd 455 . . . . . . . . 9 (((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ π‘₯ ∈ 𝑗 ∧ 𝑦 ∈ π‘₯) ∧ (𝑒 ∈ 𝑗 ∧ (𝑒 βŠ† π‘₯ ∧ 𝑦 ∈ 𝑒 ∧ (𝑗 β†Ύt 𝑒) ∈ 𝑛-Locally 𝐴))) β†’ ((𝑣 ∈ 𝒫 𝑒 ∧ βˆƒπ‘§ ∈ (𝑗 β†Ύt 𝑒)(𝑦 ∈ 𝑧 ∧ 𝑧 βŠ† 𝑣 ∧ ((𝑗 β†Ύt 𝑒) β†Ύt 𝑣) ∈ 𝐴)) β†’ (𝑣 ∈ (((neiβ€˜π‘—)β€˜{𝑦}) ∩ 𝒫 π‘₯) ∧ (𝑗 β†Ύt 𝑣) ∈ 𝐴)))
4847reximdv2 3158 . . . . . . . 8 (((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ π‘₯ ∈ 𝑗 ∧ 𝑦 ∈ π‘₯) ∧ (𝑒 ∈ 𝑗 ∧ (𝑒 βŠ† π‘₯ ∧ 𝑦 ∈ 𝑒 ∧ (𝑗 β†Ύt 𝑒) ∈ 𝑛-Locally 𝐴))) β†’ (βˆƒπ‘£ ∈ 𝒫 π‘’βˆƒπ‘§ ∈ (𝑗 β†Ύt 𝑒)(𝑦 ∈ 𝑧 ∧ 𝑧 βŠ† 𝑣 ∧ ((𝑗 β†Ύt 𝑒) β†Ύt 𝑣) ∈ 𝐴) β†’ βˆƒπ‘£ ∈ (((neiβ€˜π‘—)β€˜{𝑦}) ∩ 𝒫 π‘₯)(𝑗 β†Ύt 𝑣) ∈ 𝐴))
4913, 48mpd 15 . . . . . . 7 (((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ π‘₯ ∈ 𝑗 ∧ 𝑦 ∈ π‘₯) ∧ (𝑒 ∈ 𝑗 ∧ (𝑒 βŠ† π‘₯ ∧ 𝑦 ∈ 𝑒 ∧ (𝑗 β†Ύt 𝑒) ∈ 𝑛-Locally 𝐴))) β†’ βˆƒπ‘£ ∈ (((neiβ€˜π‘—)β€˜{𝑦}) ∩ 𝒫 π‘₯)(𝑗 β†Ύt 𝑣) ∈ 𝐴)
502, 49rexlimddv 3155 . . . . . 6 ((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ π‘₯ ∈ 𝑗 ∧ 𝑦 ∈ π‘₯) β†’ βˆƒπ‘£ ∈ (((neiβ€˜π‘—)β€˜{𝑦}) ∩ 𝒫 π‘₯)(𝑗 β†Ύt 𝑣) ∈ 𝐴)
51503expb 1121 . . . . 5 ((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ (π‘₯ ∈ 𝑗 ∧ 𝑦 ∈ π‘₯)) β†’ βˆƒπ‘£ ∈ (((neiβ€˜π‘—)β€˜{𝑦}) ∩ 𝒫 π‘₯)(𝑗 β†Ύt 𝑣) ∈ 𝐴)
5251ralrimivva 3194 . . . 4 (𝑗 ∈ Locally 𝑛-Locally 𝐴 β†’ βˆ€π‘₯ ∈ 𝑗 βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘£ ∈ (((neiβ€˜π‘—)β€˜{𝑦}) ∩ 𝒫 π‘₯)(𝑗 β†Ύt 𝑣) ∈ 𝐴)
53 isnlly 22843 . . . 4 (𝑗 ∈ 𝑛-Locally 𝐴 ↔ (𝑗 ∈ Top ∧ βˆ€π‘₯ ∈ 𝑗 βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘£ ∈ (((neiβ€˜π‘—)β€˜{𝑦}) ∩ 𝒫 π‘₯)(𝑗 β†Ύt 𝑣) ∈ 𝐴))
541, 52, 53sylanbrc 584 . . 3 (𝑗 ∈ Locally 𝑛-Locally 𝐴 β†’ 𝑗 ∈ 𝑛-Locally 𝐴)
5554ssriv 3952 . 2 Locally 𝑛-Locally 𝐴 βŠ† 𝑛-Locally 𝐴
56 nllyrest 22860 . . . . 5 ((𝑗 ∈ 𝑛-Locally 𝐴 ∧ π‘₯ ∈ 𝑗) β†’ (𝑗 β†Ύt π‘₯) ∈ 𝑛-Locally 𝐴)
5756adantl 483 . . . 4 ((⊀ ∧ (𝑗 ∈ 𝑛-Locally 𝐴 ∧ π‘₯ ∈ 𝑗)) β†’ (𝑗 β†Ύt π‘₯) ∈ 𝑛-Locally 𝐴)
58 nllytop 22847 . . . . . 6 (𝑗 ∈ 𝑛-Locally 𝐴 β†’ 𝑗 ∈ Top)
5958ssriv 3952 . . . . 5 𝑛-Locally 𝐴 βŠ† Top
6059a1i 11 . . . 4 (⊀ β†’ 𝑛-Locally 𝐴 βŠ† Top)
6157, 60restlly 22857 . . 3 (⊀ β†’ 𝑛-Locally 𝐴 βŠ† Locally 𝑛-Locally 𝐴)
6261mptru 1549 . 2 𝑛-Locally 𝐴 βŠ† Locally 𝑛-Locally 𝐴
6355, 62eqssi 3964 1 Locally 𝑛-Locally 𝐴 = 𝑛-Locally 𝐴
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542  βŠ€wtru 1543   ∈ wcel 2107  βˆ€wral 3061  βˆƒwrex 3070   ∩ cin 3913   βŠ† wss 3914  π’« cpw 4564  {csn 4590  βˆͺ cuni 4869  β€˜cfv 6500  (class class class)co 7361   β†Ύt crest 17310  Topctop 22265  neicnei 22471  Locally clly 22838  π‘›-Locally cnlly 22839
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 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
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 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-en 8890  df-fin 8893  df-fi 9355  df-rest 17312  df-topgen 17333  df-top 22266  df-topon 22283  df-bases 22319  df-nei 22472  df-lly 22840  df-nlly 22841
This theorem is referenced by: (None)
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