MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nllyidm Structured version   Visualization version   GIF version
Theorem nllyidm 22992
Description: Idempotence of the "n-locally" predicate, i.e. being "n-locally 𝐴 " is a local property. (Use loclly 22990 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 22975 . . . 4 (𝑗 ∈ Locally 𝑛-Locally 𝐴 β†’ 𝑗 ∈ Top)
2 llyi 22977 . . . . . . 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 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†’ (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 4005 . . . . . . . . . 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 1191 . . . . . . . . . . . 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 22680 . . . . . . . . . . 11 ((𝑗 ∈ Top ∧ 𝑒 ∈ 𝑗) β†’ (𝑒 ∈ (𝑗 β†Ύt 𝑒) ↔ (𝑒 ∈ 𝑗 ∧ 𝑒 βŠ† 𝑒)))
97, 4, 8syl2anc 584 . . . . . . . . . 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 1222 . . . . . . . . 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 22979 . . . . . . . . 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 1371 . . . . . . . 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 22680 . . . . . . . . . . . . . 14 ((𝑗 ∈ Top ∧ 𝑒 ∈ 𝑗) β†’ (𝑧 ∈ (𝑗 β†Ύt 𝑒) ↔ (𝑧 ∈ 𝑗 ∧ 𝑧 βŠ† 𝑒)))
157, 4, 14syl2anc 584 . . . . . . . . . . . . 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 481 . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . 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 1232 . . . . . . . . . . . . . . . . . 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 1263 . . . . . . . . . . . . . . . . . 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 22622 . . . . . . . . . . . . . . . . . 18 ((𝑗 ∈ Top ∧ 𝑧 ∈ 𝑗 ∧ 𝑦 ∈ 𝑧) β†’ 𝑧 ∈ ((neiβ€˜π‘—)β€˜{𝑦}))
2117, 18, 19, 20syl3anc 1371 . . . . . . . . . . . . . . . . 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 1264 . . . . . . . . . . . . . . . . 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 1194 . . . . . . . . . . . . . . . . . . 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 4611 . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . 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 4941 . . . . . . . . . . . . . . . . . . 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 3992 . . . . . . . . . . . . . . . . 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 2732 . . . . . . . . . . . . . . . . . 18 βˆͺ 𝑗 = βˆͺ 𝑗
3029ssnei2 22619 . . . . . . . . . . . . . . . . 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 1221 . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . 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 3992 . . . . . . . . . . . . . . . . 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 4607 . . . . . . . . . . . . . . . . 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 22668 . . . . . . . . . . . . . . . . 17 ((𝑗 ∈ Top ∧ 𝑣 βŠ† 𝑒 ∧ 𝑒 ∈ 𝑗) β†’ ((𝑗 β†Ύt 𝑒) β†Ύt 𝑣) = (𝑗 β†Ύt 𝑣))
3917, 24, 25, 38syl3anc 1371 . . . . . . . . . . . . . . . 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 1265 . . . . . . . . . . . . . . . 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 2834 . . . . . . . . . . . . . . 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 512 . . . . . . . . . . . . . 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 1354 . . . . . . . . . . . . 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 407 . . . . . . . . . . . 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 3153 . . . . . . . . . 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 454 . . . . . . . . 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 3164 . . . . . . . 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 3161 . . . . . 6 ((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ π‘₯ ∈ 𝑗 ∧ 𝑦 ∈ π‘₯) β†’ βˆƒπ‘£ ∈ (((neiβ€˜π‘—)β€˜{𝑦}) ∩ 𝒫 π‘₯)(𝑗 β†Ύt 𝑣) ∈ 𝐴)
51503expb 1120 . . . . 5 ((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ (π‘₯ ∈ 𝑗 ∧ 𝑦 ∈ π‘₯)) β†’ βˆƒπ‘£ ∈ (((neiβ€˜π‘—)β€˜{𝑦}) ∩ 𝒫 π‘₯)(𝑗 β†Ύt 𝑣) ∈ 𝐴)
5251ralrimivva 3200 . . . 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 22972 . . . 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 583 . . 3 (𝑗 ∈ Locally 𝑛-Locally 𝐴 β†’ 𝑗 ∈ 𝑛-Locally 𝐴)
5554ssriv 3986 . 2 Locally 𝑛-Locally 𝐴 βŠ† 𝑛-Locally 𝐴
56 nllyrest 22989 . . . . 5 ((𝑗 ∈ 𝑛-Locally 𝐴 ∧ π‘₯ ∈ 𝑗) β†’ (𝑗 β†Ύt π‘₯) ∈ 𝑛-Locally 𝐴)
5756adantl 482 . . . 4 ((⊀ ∧ (𝑗 ∈ 𝑛-Locally 𝐴 ∧ π‘₯ ∈ 𝑗)) β†’ (𝑗 β†Ύt π‘₯) ∈ 𝑛-Locally 𝐴)
58 nllytop 22976 . . . . . 6 (𝑗 ∈ 𝑛-Locally 𝐴 β†’ 𝑗 ∈ Top)
5958ssriv 3986 . . . . 5 𝑛-Locally 𝐴 βŠ† Top
6059a1i 11 . . . 4 (⊀ β†’ 𝑛-Locally 𝐴 βŠ† Top)
6157, 60restlly 22986 . . 3 (⊀ β†’ 𝑛-Locally 𝐴 βŠ† Locally 𝑛-Locally 𝐴)
6261mptru 1548 . 2 𝑛-Locally 𝐴 βŠ† Locally 𝑛-Locally 𝐴
6355, 62eqssi 3998 1 Locally 𝑛-Locally 𝐴 = 𝑛-Locally 𝐴
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541  βŠ€wtru 1542   ∈ wcel 2106  βˆ€wral 3061  βˆƒwrex 3070   ∩ cin 3947   βŠ† wss 3948  π’« cpw 4602  {csn 4628  βˆͺ cuni 4908  β€˜cfv 6543  (class class class)co 7408   β†Ύt crest 17365  Topctop 22394  neicnei 22600  Locally clly 22967  π‘›-Locally cnlly 22968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-en 8939  df-fin 8942  df-fi 9405  df-rest 17367  df-topgen 17388  df-top 22395  df-topon 22412  df-bases 22448  df-nei 22601  df-lly 22969  df-nlly 22970
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator