Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ntrneiel2 Structured version   Visualization version   GIF version

Theorem ntrneiel2 42919
Description: Membership in iterated interior of a set is equivalent to there existing a particular neighborhood of that member such that points are members of that neighborhood if and only if the set is a neighborhood of each of those points. (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)
ntrneiel2.x (πœ‘ β†’ 𝑋 ∈ 𝐡)
ntrneiel2.s (πœ‘ β†’ 𝑆 ∈ 𝒫 𝐡)
Assertion
Ref Expression
ntrneiel2 (πœ‘ β†’ (𝑋 ∈ (πΌβ€˜(πΌβ€˜π‘†)) ↔ βˆƒπ‘’ ∈ (π‘β€˜π‘‹)βˆ€π‘¦ ∈ 𝐡 (𝑦 ∈ 𝑒 ↔ 𝑆 ∈ (π‘β€˜π‘¦))))
Distinct variable groups:   𝐡,𝑖,𝑗,π‘˜,𝑙,π‘š,𝑦   𝑒,𝐡,𝑦   π‘˜,𝐼,𝑙,π‘š,𝑦   𝑒,𝑁,𝑦   𝑆,π‘š,𝑦   𝑒,𝑆   𝑋,𝑙,π‘š,𝑦   𝑒,𝑋   πœ‘,𝑖,𝑗,π‘˜,𝑙,𝑦   πœ‘,𝑒
Allowed substitution hints:   πœ‘(π‘š)   𝑆(𝑖,𝑗,π‘˜,𝑙)   𝐹(𝑦,𝑒,𝑖,𝑗,π‘˜,π‘š,𝑙)   𝐼(𝑒,𝑖,𝑗)   𝑁(𝑖,𝑗,π‘˜,π‘š,𝑙)   𝑂(𝑦,𝑒,𝑖,𝑗,π‘˜,π‘š,𝑙)   𝑋(𝑖,𝑗,π‘˜)

Proof of Theorem ntrneiel2
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)
4 ntrneiel2.x . . 3 (πœ‘ β†’ 𝑋 ∈ 𝐡)
51, 2, 3ntrneiiex 42909 . . . . 5 (πœ‘ β†’ 𝐼 ∈ (𝒫 𝐡 ↑m 𝒫 𝐡))
6 elmapi 8845 . . . . 5 (𝐼 ∈ (𝒫 𝐡 ↑m 𝒫 𝐡) β†’ 𝐼:𝒫 π΅βŸΆπ’« 𝐡)
75, 6syl 17 . . . 4 (πœ‘ β†’ 𝐼:𝒫 π΅βŸΆπ’« 𝐡)
8 ntrneiel2.s . . . 4 (πœ‘ β†’ 𝑆 ∈ 𝒫 𝐡)
97, 8ffvelcdmd 7087 . . 3 (πœ‘ β†’ (πΌβ€˜π‘†) ∈ 𝒫 𝐡)
101, 2, 3, 4, 9ntrneiel 42914 . 2 (πœ‘ β†’ (𝑋 ∈ (πΌβ€˜(πΌβ€˜π‘†)) ↔ (πΌβ€˜π‘†) ∈ (π‘β€˜π‘‹)))
111, 2, 3, 8ntrneifv4 42918 . . . 4 (πœ‘ β†’ (πΌβ€˜π‘†) = {𝑦 ∈ 𝐡 ∣ 𝑆 ∈ (π‘β€˜π‘¦)})
12 df-rab 3433 . . . 4 {𝑦 ∈ 𝐡 ∣ 𝑆 ∈ (π‘β€˜π‘¦)} = {𝑦 ∣ (𝑦 ∈ 𝐡 ∧ 𝑆 ∈ (π‘β€˜π‘¦))}
1311, 12eqtrdi 2788 . . 3 (πœ‘ β†’ (πΌβ€˜π‘†) = {𝑦 ∣ (𝑦 ∈ 𝐡 ∧ 𝑆 ∈ (π‘β€˜π‘¦))})
1413eleq1d 2818 . 2 (πœ‘ β†’ ((πΌβ€˜π‘†) ∈ (π‘β€˜π‘‹) ↔ {𝑦 ∣ (𝑦 ∈ 𝐡 ∧ 𝑆 ∈ (π‘β€˜π‘¦))} ∈ (π‘β€˜π‘‹)))
15 clabel 2881 . . . 4 ({𝑦 ∣ (𝑦 ∈ 𝐡 ∧ 𝑆 ∈ (π‘β€˜π‘¦))} ∈ (π‘β€˜π‘‹) ↔ βˆƒπ‘’(𝑒 ∈ (π‘β€˜π‘‹) ∧ βˆ€π‘¦(𝑦 ∈ 𝑒 ↔ (𝑦 ∈ 𝐡 ∧ 𝑆 ∈ (π‘β€˜π‘¦)))))
16 df-rex 3071 . . . 4 (βˆƒπ‘’ ∈ (π‘β€˜π‘‹)βˆ€π‘¦(𝑦 ∈ 𝑒 ↔ (𝑦 ∈ 𝐡 ∧ 𝑆 ∈ (π‘β€˜π‘¦))) ↔ βˆƒπ‘’(𝑒 ∈ (π‘β€˜π‘‹) ∧ βˆ€π‘¦(𝑦 ∈ 𝑒 ↔ (𝑦 ∈ 𝐡 ∧ 𝑆 ∈ (π‘β€˜π‘¦)))))
1715, 16bitr4i 277 . . 3 ({𝑦 ∣ (𝑦 ∈ 𝐡 ∧ 𝑆 ∈ (π‘β€˜π‘¦))} ∈ (π‘β€˜π‘‹) ↔ βˆƒπ‘’ ∈ (π‘β€˜π‘‹)βˆ€π‘¦(𝑦 ∈ 𝑒 ↔ (𝑦 ∈ 𝐡 ∧ 𝑆 ∈ (π‘β€˜π‘¦))))
18 ibar 529 . . . . . . . 8 (𝑦 ∈ 𝐡 β†’ (𝑆 ∈ (π‘β€˜π‘¦) ↔ (𝑦 ∈ 𝐡 ∧ 𝑆 ∈ (π‘β€˜π‘¦))))
1918bibi2d 342 . . . . . . 7 (𝑦 ∈ 𝐡 β†’ ((𝑦 ∈ 𝑒 ↔ 𝑆 ∈ (π‘β€˜π‘¦)) ↔ (𝑦 ∈ 𝑒 ↔ (𝑦 ∈ 𝐡 ∧ 𝑆 ∈ (π‘β€˜π‘¦)))))
2019ralbiia 3091 . . . . . 6 (βˆ€π‘¦ ∈ 𝐡 (𝑦 ∈ 𝑒 ↔ 𝑆 ∈ (π‘β€˜π‘¦)) ↔ βˆ€π‘¦ ∈ 𝐡 (𝑦 ∈ 𝑒 ↔ (𝑦 ∈ 𝐡 ∧ 𝑆 ∈ (π‘β€˜π‘¦))))
21 ssv 4006 . . . . . . . 8 𝐡 βŠ† V
2221a1i 11 . . . . . . 7 ((πœ‘ ∧ 𝑒 ∈ (π‘β€˜π‘‹)) β†’ 𝐡 βŠ† V)
23 vex 3478 . . . . . . . . . 10 𝑦 ∈ V
24 eldif 3958 . . . . . . . . . 10 (𝑦 ∈ (V βˆ– 𝐡) ↔ (𝑦 ∈ V ∧ Β¬ 𝑦 ∈ 𝐡))
2523, 24mpbiran 707 . . . . . . . . 9 (𝑦 ∈ (V βˆ– 𝐡) ↔ Β¬ 𝑦 ∈ 𝐡)
261, 2, 3ntrneinex 42910 . . . . . . . . . . . . . . . . . 18 (πœ‘ β†’ 𝑁 ∈ (𝒫 𝒫 𝐡 ↑m 𝐡))
27 elmapi 8845 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ (𝒫 𝒫 𝐡 ↑m 𝐡) β†’ 𝑁:π΅βŸΆπ’« 𝒫 𝐡)
2826, 27syl 17 . . . . . . . . . . . . . . . . 17 (πœ‘ β†’ 𝑁:π΅βŸΆπ’« 𝒫 𝐡)
2928, 4ffvelcdmd 7087 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ (π‘β€˜π‘‹) ∈ 𝒫 𝒫 𝐡)
3029elpwid 4611 . . . . . . . . . . . . . . 15 (πœ‘ β†’ (π‘β€˜π‘‹) βŠ† 𝒫 𝐡)
3130sselda 3982 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 𝑒 ∈ (π‘β€˜π‘‹)) β†’ 𝑒 ∈ 𝒫 𝐡)
3231elpwid 4611 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑒 ∈ (π‘β€˜π‘‹)) β†’ 𝑒 βŠ† 𝐡)
3332sseld 3981 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑒 ∈ (π‘β€˜π‘‹)) β†’ (𝑦 ∈ 𝑒 β†’ 𝑦 ∈ 𝐡))
3433con3dimp 409 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑒 ∈ (π‘β€˜π‘‹)) ∧ Β¬ 𝑦 ∈ 𝐡) β†’ Β¬ 𝑦 ∈ 𝑒)
35 pm3.14 994 . . . . . . . . . . . . 13 ((Β¬ 𝑦 ∈ 𝐡 ∨ Β¬ 𝑆 ∈ (π‘β€˜π‘¦)) β†’ Β¬ (𝑦 ∈ 𝐡 ∧ 𝑆 ∈ (π‘β€˜π‘¦)))
3635orcs 873 . . . . . . . . . . . 12 (Β¬ 𝑦 ∈ 𝐡 β†’ Β¬ (𝑦 ∈ 𝐡 ∧ 𝑆 ∈ (π‘β€˜π‘¦)))
3736adantl 482 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑒 ∈ (π‘β€˜π‘‹)) ∧ Β¬ 𝑦 ∈ 𝐡) β†’ Β¬ (𝑦 ∈ 𝐡 ∧ 𝑆 ∈ (π‘β€˜π‘¦)))
3834, 372falsed 376 . . . . . . . . . 10 (((πœ‘ ∧ 𝑒 ∈ (π‘β€˜π‘‹)) ∧ Β¬ 𝑦 ∈ 𝐡) β†’ (𝑦 ∈ 𝑒 ↔ (𝑦 ∈ 𝐡 ∧ 𝑆 ∈ (π‘β€˜π‘¦))))
3938ex 413 . . . . . . . . 9 ((πœ‘ ∧ 𝑒 ∈ (π‘β€˜π‘‹)) β†’ (Β¬ 𝑦 ∈ 𝐡 β†’ (𝑦 ∈ 𝑒 ↔ (𝑦 ∈ 𝐡 ∧ 𝑆 ∈ (π‘β€˜π‘¦)))))
4025, 39biimtrid 241 . . . . . . . 8 ((πœ‘ ∧ 𝑒 ∈ (π‘β€˜π‘‹)) β†’ (𝑦 ∈ (V βˆ– 𝐡) β†’ (𝑦 ∈ 𝑒 ↔ (𝑦 ∈ 𝐡 ∧ 𝑆 ∈ (π‘β€˜π‘¦)))))
4140ralrimiv 3145 . . . . . . 7 ((πœ‘ ∧ 𝑒 ∈ (π‘β€˜π‘‹)) β†’ βˆ€π‘¦ ∈ (V βˆ– 𝐡)(𝑦 ∈ 𝑒 ↔ (𝑦 ∈ 𝐡 ∧ 𝑆 ∈ (π‘β€˜π‘¦))))
4222, 41raldifeq 4493 . . . . . 6 ((πœ‘ ∧ 𝑒 ∈ (π‘β€˜π‘‹)) β†’ (βˆ€π‘¦ ∈ 𝐡 (𝑦 ∈ 𝑒 ↔ (𝑦 ∈ 𝐡 ∧ 𝑆 ∈ (π‘β€˜π‘¦))) ↔ βˆ€π‘¦ ∈ V (𝑦 ∈ 𝑒 ↔ (𝑦 ∈ 𝐡 ∧ 𝑆 ∈ (π‘β€˜π‘¦)))))
4320, 42bitrid 282 . . . . 5 ((πœ‘ ∧ 𝑒 ∈ (π‘β€˜π‘‹)) β†’ (βˆ€π‘¦ ∈ 𝐡 (𝑦 ∈ 𝑒 ↔ 𝑆 ∈ (π‘β€˜π‘¦)) ↔ βˆ€π‘¦ ∈ V (𝑦 ∈ 𝑒 ↔ (𝑦 ∈ 𝐡 ∧ 𝑆 ∈ (π‘β€˜π‘¦)))))
44 ralv 3498 . . . . 5 (βˆ€π‘¦ ∈ V (𝑦 ∈ 𝑒 ↔ (𝑦 ∈ 𝐡 ∧ 𝑆 ∈ (π‘β€˜π‘¦))) ↔ βˆ€π‘¦(𝑦 ∈ 𝑒 ↔ (𝑦 ∈ 𝐡 ∧ 𝑆 ∈ (π‘β€˜π‘¦))))
4543, 44bitr2di 287 . . . 4 ((πœ‘ ∧ 𝑒 ∈ (π‘β€˜π‘‹)) β†’ (βˆ€π‘¦(𝑦 ∈ 𝑒 ↔ (𝑦 ∈ 𝐡 ∧ 𝑆 ∈ (π‘β€˜π‘¦))) ↔ βˆ€π‘¦ ∈ 𝐡 (𝑦 ∈ 𝑒 ↔ 𝑆 ∈ (π‘β€˜π‘¦))))
4645rexbidva 3176 . . 3 (πœ‘ β†’ (βˆƒπ‘’ ∈ (π‘β€˜π‘‹)βˆ€π‘¦(𝑦 ∈ 𝑒 ↔ (𝑦 ∈ 𝐡 ∧ 𝑆 ∈ (π‘β€˜π‘¦))) ↔ βˆƒπ‘’ ∈ (π‘β€˜π‘‹)βˆ€π‘¦ ∈ 𝐡 (𝑦 ∈ 𝑒 ↔ 𝑆 ∈ (π‘β€˜π‘¦))))
4717, 46bitrid 282 . 2 (πœ‘ β†’ ({𝑦 ∣ (𝑦 ∈ 𝐡 ∧ 𝑆 ∈ (π‘β€˜π‘¦))} ∈ (π‘β€˜π‘‹) ↔ βˆƒπ‘’ ∈ (π‘β€˜π‘‹)βˆ€π‘¦ ∈ 𝐡 (𝑦 ∈ 𝑒 ↔ 𝑆 ∈ (π‘β€˜π‘¦))))
4810, 14, 473bitrd 304 1 (πœ‘ β†’ (𝑋 ∈ (πΌβ€˜(πΌβ€˜π‘†)) ↔ βˆƒπ‘’ ∈ (π‘β€˜π‘‹)βˆ€π‘¦ ∈ 𝐡 (𝑦 ∈ 𝑒 ↔ 𝑆 ∈ (π‘β€˜π‘¦))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396  βˆ€wal 1539   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106  {cab 2709  βˆ€wral 3061  βˆƒwrex 3070  {crab 3432  Vcvv 3474   βˆ– cdif 3945   βŠ† wss 3948  π’« cpw 4602   class class class wbr 5148   ↦ cmpt 5231  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7411   ∈ cmpo 7413   ↑m cmap 8822
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 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  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-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  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-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 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-map 8824
This theorem is referenced by:  ntrneik4  42934
  Copyright terms: Public domain W3C validator