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Theorem ntrneik4 43428
Description: Idempotence of the interior function is equivalent to stating a set, 𝑠, is a neighborhood of a point, π‘₯ is equivalent to there existing a special neighborhood, 𝑒, of π‘₯ such that a point is an element of the special neighborhood if and only if 𝑠 is also a neighborhood of the point. (Contributed by RP, 11-Jul-2021.)
Hypotheses
Ref Expression
ntrnei.o 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (π‘˜ ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {π‘š ∈ 𝑖 ∣ 𝑙 ∈ (π‘˜β€˜π‘š)})))
ntrnei.f 𝐹 = (𝒫 𝐡𝑂𝐡)
ntrnei.r (πœ‘ β†’ 𝐼𝐹𝑁)
Assertion
Ref Expression
ntrneik4 (πœ‘ β†’ (βˆ€π‘  ∈ 𝒫 𝐡(πΌβ€˜(πΌβ€˜π‘ )) = (πΌβ€˜π‘ ) ↔ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘  ∈ 𝒫 𝐡(𝑠 ∈ (π‘β€˜π‘₯) ↔ βˆƒπ‘’ ∈ (π‘β€˜π‘₯)βˆ€π‘¦ ∈ 𝐡 (𝑦 ∈ 𝑒 ↔ 𝑠 ∈ (π‘β€˜π‘¦)))))
Distinct variable groups:   𝐡,𝑖,𝑗,π‘˜,𝑙,π‘š,𝑠,π‘₯,𝑦   π‘˜,𝐼,𝑙,π‘š,π‘₯,𝑦   πœ‘,𝑖,𝑗,π‘˜,𝑙,𝑠,π‘₯   𝑒,𝐡,𝑠,π‘₯,𝑦   𝑒,𝑁,𝑦   πœ‘,𝑒,𝑦
Allowed substitution hints:   πœ‘(π‘š)   𝐹(π‘₯,𝑦,𝑒,𝑖,𝑗,π‘˜,π‘š,𝑠,𝑙)   𝐼(𝑒,𝑖,𝑗,𝑠)   𝑁(π‘₯,𝑖,𝑗,π‘˜,π‘š,𝑠,𝑙)   𝑂(π‘₯,𝑦,𝑒,𝑖,𝑗,π‘˜,π‘š,𝑠,𝑙)

Proof of Theorem ntrneik4
StepHypRef Expression
1 ntrnei.o . . 3 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (π‘˜ ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {π‘š ∈ 𝑖 ∣ 𝑙 ∈ (π‘˜β€˜π‘š)})))
2 ntrnei.f . . 3 𝐹 = (𝒫 𝐡𝑂𝐡)
3 ntrnei.r . . 3 (πœ‘ β†’ 𝐼𝐹𝑁)
41, 2, 3ntrneik4w 43427 . 2 (πœ‘ β†’ (βˆ€π‘  ∈ 𝒫 𝐡(πΌβ€˜(πΌβ€˜π‘ )) = (πΌβ€˜π‘ ) ↔ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘  ∈ 𝒫 𝐡(𝑠 ∈ (π‘β€˜π‘₯) ↔ (πΌβ€˜π‘ ) ∈ (π‘β€˜π‘₯))))
53ad2antrr 723 . . . . . . 7 (((πœ‘ ∧ π‘₯ ∈ 𝐡) ∧ 𝑠 ∈ 𝒫 𝐡) β†’ 𝐼𝐹𝑁)
6 simplr 766 . . . . . . 7 (((πœ‘ ∧ π‘₯ ∈ 𝐡) ∧ 𝑠 ∈ 𝒫 𝐡) β†’ π‘₯ ∈ 𝐡)
71, 2, 3ntrneiiex 43403 . . . . . . . . . 10 (πœ‘ β†’ 𝐼 ∈ (𝒫 𝐡 ↑m 𝒫 𝐡))
8 elmapi 8845 . . . . . . . . . 10 (𝐼 ∈ (𝒫 𝐡 ↑m 𝒫 𝐡) β†’ 𝐼:𝒫 π΅βŸΆπ’« 𝐡)
97, 8syl 17 . . . . . . . . 9 (πœ‘ β†’ 𝐼:𝒫 π΅βŸΆπ’« 𝐡)
109ffvelcdmda 7080 . . . . . . . 8 ((πœ‘ ∧ 𝑠 ∈ 𝒫 𝐡) β†’ (πΌβ€˜π‘ ) ∈ 𝒫 𝐡)
1110adantlr 712 . . . . . . 7 (((πœ‘ ∧ π‘₯ ∈ 𝐡) ∧ 𝑠 ∈ 𝒫 𝐡) β†’ (πΌβ€˜π‘ ) ∈ 𝒫 𝐡)
121, 2, 5, 6, 11ntrneiel 43408 . . . . . 6 (((πœ‘ ∧ π‘₯ ∈ 𝐡) ∧ 𝑠 ∈ 𝒫 𝐡) β†’ (π‘₯ ∈ (πΌβ€˜(πΌβ€˜π‘ )) ↔ (πΌβ€˜π‘ ) ∈ (π‘β€˜π‘₯)))
13 simpr 484 . . . . . . 7 (((πœ‘ ∧ π‘₯ ∈ 𝐡) ∧ 𝑠 ∈ 𝒫 𝐡) β†’ 𝑠 ∈ 𝒫 𝐡)
141, 2, 5, 6, 13ntrneiel2 43413 . . . . . 6 (((πœ‘ ∧ π‘₯ ∈ 𝐡) ∧ 𝑠 ∈ 𝒫 𝐡) β†’ (π‘₯ ∈ (πΌβ€˜(πΌβ€˜π‘ )) ↔ βˆƒπ‘’ ∈ (π‘β€˜π‘₯)βˆ€π‘¦ ∈ 𝐡 (𝑦 ∈ 𝑒 ↔ 𝑠 ∈ (π‘β€˜π‘¦))))
1512, 14bitr3d 281 . . . . 5 (((πœ‘ ∧ π‘₯ ∈ 𝐡) ∧ 𝑠 ∈ 𝒫 𝐡) β†’ ((πΌβ€˜π‘ ) ∈ (π‘β€˜π‘₯) ↔ βˆƒπ‘’ ∈ (π‘β€˜π‘₯)βˆ€π‘¦ ∈ 𝐡 (𝑦 ∈ 𝑒 ↔ 𝑠 ∈ (π‘β€˜π‘¦))))
1615bibi2d 342 . . . 4 (((πœ‘ ∧ π‘₯ ∈ 𝐡) ∧ 𝑠 ∈ 𝒫 𝐡) β†’ ((𝑠 ∈ (π‘β€˜π‘₯) ↔ (πΌβ€˜π‘ ) ∈ (π‘β€˜π‘₯)) ↔ (𝑠 ∈ (π‘β€˜π‘₯) ↔ βˆƒπ‘’ ∈ (π‘β€˜π‘₯)βˆ€π‘¦ ∈ 𝐡 (𝑦 ∈ 𝑒 ↔ 𝑠 ∈ (π‘β€˜π‘¦)))))
1716ralbidva 3169 . . 3 ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ (βˆ€π‘  ∈ 𝒫 𝐡(𝑠 ∈ (π‘β€˜π‘₯) ↔ (πΌβ€˜π‘ ) ∈ (π‘β€˜π‘₯)) ↔ βˆ€π‘  ∈ 𝒫 𝐡(𝑠 ∈ (π‘β€˜π‘₯) ↔ βˆƒπ‘’ ∈ (π‘β€˜π‘₯)βˆ€π‘¦ ∈ 𝐡 (𝑦 ∈ 𝑒 ↔ 𝑠 ∈ (π‘β€˜π‘¦)))))
1817ralbidva 3169 . 2 (πœ‘ β†’ (βˆ€π‘₯ ∈ 𝐡 βˆ€π‘  ∈ 𝒫 𝐡(𝑠 ∈ (π‘β€˜π‘₯) ↔ (πΌβ€˜π‘ ) ∈ (π‘β€˜π‘₯)) ↔ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘  ∈ 𝒫 𝐡(𝑠 ∈ (π‘β€˜π‘₯) ↔ βˆƒπ‘’ ∈ (π‘β€˜π‘₯)βˆ€π‘¦ ∈ 𝐡 (𝑦 ∈ 𝑒 ↔ 𝑠 ∈ (π‘β€˜π‘¦)))))
194, 18bitrd 279 1 (πœ‘ β†’ (βˆ€π‘  ∈ 𝒫 𝐡(πΌβ€˜(πΌβ€˜π‘ )) = (πΌβ€˜π‘ ) ↔ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘  ∈ 𝒫 𝐡(𝑠 ∈ (π‘β€˜π‘₯) ↔ βˆƒπ‘’ ∈ (π‘β€˜π‘₯)βˆ€π‘¦ ∈ 𝐡 (𝑦 ∈ 𝑒 ↔ 𝑠 ∈ (π‘β€˜π‘¦)))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055  βˆƒwrex 3064  {crab 3426  Vcvv 3468  π’« cpw 4597   class class class wbr 5141   ↦ cmpt 5224  βŸΆwf 6533  β€˜cfv 6537  (class class class)co 7405   ∈ cmpo 7407   ↑m cmap 8822
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 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  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-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-map 8824
This theorem is referenced by: (None)
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