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Theorem nllyidm 23344
Description: Idempotence of the "n-locally" predicate, i.e. being "n-locally 𝐴 " is a local property. (Use loclly 23342 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 23327 . . . 4 (𝑗 ∈ Locally 𝑛-Locally 𝐴 β†’ 𝑗 ∈ Top)
2 llyi 23329 . . . . . . 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 1220 . . . . . . . . 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 768 . . . . . . . . . 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 4000 . . . . . . . . . 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 1188 . . . . . . . . . . . 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 23032 . . . . . . . . . . 11 ((𝑗 ∈ Top ∧ 𝑒 ∈ 𝑗) β†’ (𝑒 ∈ (𝑗 β†Ύt 𝑒) ↔ (𝑒 ∈ 𝑗 ∧ 𝑒 βŠ† 𝑒)))
97, 4, 8syl2anc 583 . . . . . . . . . 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 710 . . . . . . . . 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 1219 . . . . . . . . 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 23331 . . . . . . . . 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 1368 . . . . . . . 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 23032 . . . . . . . . . . . . . 14 ((𝑗 ∈ Top ∧ 𝑒 ∈ 𝑗) β†’ (𝑧 ∈ (𝑗 β†Ύt 𝑒) ↔ (𝑧 ∈ 𝑗 ∧ 𝑧 βŠ† 𝑒)))
157, 4, 14syl2anc 583 . . . . . . . . . . . . 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 480 . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . 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 1229 . . . . . . . . . . . . . . . . . 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 1260 . . . . . . . . . . . . . . . . . 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 22974 . . . . . . . . . . . . . . . . . 18 ((𝑗 ∈ Top ∧ 𝑧 ∈ 𝑗 ∧ 𝑦 ∈ 𝑧) β†’ 𝑧 ∈ ((neiβ€˜π‘—)β€˜{𝑦}))
2117, 18, 19, 20syl3anc 1368 . . . . . . . . . . . . . . . . 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 1261 . . . . . . . . . . . . . . . . 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 1191 . . . . . . . . . . . . . . . . . . 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 4606 . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . 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 4934 . . . . . . . . . . . . . . . . . . 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 3987 . . . . . . . . . . . . . . . . 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 2726 . . . . . . . . . . . . . . . . . 18 βˆͺ 𝑗 = βˆͺ 𝑗
3029ssnei2 22971 . . . . . . . . . . . . . . . . 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 836 . . . . . . . . . . . . . . . 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 1218 . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . 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 3987 . . . . . . . . . . . . . . . . 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 4602 . . . . . . . . . . . . . . . . 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 4189 . . . . . . . . . . . . . . 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 23020 . . . . . . . . . . . . . . . . 17 ((𝑗 ∈ Top ∧ 𝑣 βŠ† 𝑒 ∧ 𝑒 ∈ 𝑗) β†’ ((𝑗 β†Ύt 𝑒) β†Ύt 𝑣) = (𝑗 β†Ύt 𝑣))
3917, 24, 25, 38syl3anc 1368 . . . . . . . . . . . . . . . 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 1262 . . . . . . . . . . . . . . . 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 2828 . . . . . . . . . . . . . . 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 511 . . . . . . . . . . . . . 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 1351 . . . . . . . . . . . . 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 406 . . . . . . . . . . . 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 453 . . . . . . . . 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 1117 . . . . 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 23324 . . . 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 582 . . 3 (𝑗 ∈ Locally 𝑛-Locally 𝐴 β†’ 𝑗 ∈ 𝑛-Locally 𝐴)
5554ssriv 3981 . 2 Locally 𝑛-Locally 𝐴 βŠ† 𝑛-Locally 𝐴
56 nllyrest 23341 . . . . 5 ((𝑗 ∈ 𝑛-Locally 𝐴 ∧ π‘₯ ∈ 𝑗) β†’ (𝑗 β†Ύt π‘₯) ∈ 𝑛-Locally 𝐴)
5756adantl 481 . . . 4 ((⊀ ∧ (𝑗 ∈ 𝑛-Locally 𝐴 ∧ π‘₯ ∈ 𝑗)) β†’ (𝑗 β†Ύt π‘₯) ∈ 𝑛-Locally 𝐴)
58 nllytop 23328 . . . . . 6 (𝑗 ∈ 𝑛-Locally 𝐴 β†’ 𝑗 ∈ Top)
5958ssriv 3981 . . . . 5 𝑛-Locally 𝐴 βŠ† Top
6059a1i 11 . . . 4 (⊀ β†’ 𝑛-Locally 𝐴 βŠ† Top)
6157, 60restlly 23338 . . 3 (⊀ β†’ 𝑛-Locally 𝐴 βŠ† Locally 𝑛-Locally 𝐴)
6261mptru 1540 . 2 𝑛-Locally 𝐴 βŠ† Locally 𝑛-Locally 𝐴
6355, 62eqssi 3993 1 Locally 𝑛-Locally 𝐴 = 𝑛-Locally 𝐴
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533  βŠ€wtru 1534   ∈ wcel 2098  βˆ€wral 3055  βˆƒwrex 3064   ∩ cin 3942   βŠ† wss 3943  π’« cpw 4597  {csn 4623  βˆͺ cuni 4902  β€˜cfv 6536  (class class class)co 7404   β†Ύt crest 17373  Topctop 22746  neicnei 22952  Locally clly 23319  π‘›-Locally cnlly 23320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-1st 7971  df-2nd 7972  df-en 8939  df-fin 8942  df-fi 9405  df-rest 17375  df-topgen 17396  df-top 22747  df-topon 22764  df-bases 22800  df-nei 22953  df-lly 23321  df-nlly 23322
This theorem is referenced by: (None)
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