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Theorem nllyidm 23411
Description: Idempotence of the "n-locally" predicate, i.e. being "n-locally 𝐴 " is a local property. (Use loclly 23409 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 23394 . . . 4 (𝑗 ∈ Locally 𝑛-Locally 𝐴 β†’ 𝑗 ∈ Top)
2 llyi 23396 . . . . . . 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 769 . . . . . . . . . 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 4003 . . . . . . . . . 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 23099 . . . . . . . . . . 11 ((𝑗 ∈ Top ∧ 𝑒 ∈ 𝑗) β†’ (𝑒 ∈ (𝑗 β†Ύt 𝑒) ↔ (𝑒 ∈ 𝑗 ∧ 𝑒 βŠ† 𝑒)))
97, 4, 8syl2anc 582 . . . . . . . . . 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 711 . . . . . . . . 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 23398 . . . . . . . . 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 23099 . . . . . . . . . . . . . 14 ((𝑗 ∈ Top ∧ 𝑒 ∈ 𝑗) β†’ (𝑧 ∈ (𝑗 β†Ύt 𝑒) ↔ (𝑧 ∈ 𝑗 ∧ 𝑧 βŠ† 𝑒)))
157, 4, 14syl2anc 582 . . . . . . . . . . . . 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 479 . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . . . 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 23041 . . . . . . . . . . . . . . . . . 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 4613 . . . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . . . . . 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 4942 . . . . . . . . . . . . . . . . . . 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 3990 . . . . . . . . . . . . . . . . 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 2727 . . . . . . . . . . . . . . . . . 18 βˆͺ 𝑗 = βˆͺ 𝑗
3029ssnei2 23038 . . . . . . . . . . . . . . . . 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 837 . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . . . . 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 3990 . . . . . . . . . . . . . . . . 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 4609 . . . . . . . . . . . . . . . . 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 4194 . . . . . . . . . . . . . . 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 23087 . . . . . . . . . . . . . . . . 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 2829 . . . . . . . . . . . . . . 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 510 . . . . . . . . . . . . . 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 405 . . . . . . . . . . . 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 3149 . . . . . . . . . 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 452 . . . . . . . . 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 3160 . . . . . . . 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 3157 . . . . . 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 3196 . . . 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 23391 . . . 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 581 . . 3 (𝑗 ∈ Locally 𝑛-Locally 𝐴 β†’ 𝑗 ∈ 𝑛-Locally 𝐴)
5554ssriv 3984 . 2 Locally 𝑛-Locally 𝐴 βŠ† 𝑛-Locally 𝐴
56 nllyrest 23408 . . . . 5 ((𝑗 ∈ 𝑛-Locally 𝐴 ∧ π‘₯ ∈ 𝑗) β†’ (𝑗 β†Ύt π‘₯) ∈ 𝑛-Locally 𝐴)
5756adantl 480 . . . 4 ((⊀ ∧ (𝑗 ∈ 𝑛-Locally 𝐴 ∧ π‘₯ ∈ 𝑗)) β†’ (𝑗 β†Ύt π‘₯) ∈ 𝑛-Locally 𝐴)
58 nllytop 23395 . . . . . 6 (𝑗 ∈ 𝑛-Locally 𝐴 β†’ 𝑗 ∈ Top)
5958ssriv 3984 . . . . 5 𝑛-Locally 𝐴 βŠ† Top
6059a1i 11 . . . 4 (⊀ β†’ 𝑛-Locally 𝐴 βŠ† Top)
6157, 60restlly 23405 . . 3 (⊀ β†’ 𝑛-Locally 𝐴 βŠ† Locally 𝑛-Locally 𝐴)
6261mptru 1540 . 2 𝑛-Locally 𝐴 βŠ† Locally 𝑛-Locally 𝐴
6355, 62eqssi 3996 1 Locally 𝑛-Locally 𝐴 = 𝑛-Locally 𝐴
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533  βŠ€wtru 1534   ∈ wcel 2098  βˆ€wral 3057  βˆƒwrex 3066   ∩ cin 3946   βŠ† wss 3947  π’« cpw 4604  {csn 4630  βˆͺ cuni 4910  β€˜cfv 6551  (class class class)co 7424   β†Ύt crest 17407  Topctop 22813  neicnei 23019  Locally clly 23386  π‘›-Locally cnlly 23387
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 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-int 4952  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-tr 5268  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5635  df-we 5637  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-ord 6375  df-on 6376  df-lim 6377  df-suc 6378  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7875  df-1st 7997  df-2nd 7998  df-en 8969  df-fin 8972  df-fi 9440  df-rest 17409  df-topgen 17430  df-top 22814  df-topon 22831  df-bases 22867  df-nei 23020  df-lly 23388  df-nlly 23389
This theorem is referenced by: (None)
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