Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ntrneifv2 Structured version   Visualization version   GIF version

Theorem ntrneifv2 37857
Description: If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then the function value of converse of 𝐹 is the interior function. (Contributed by RP, 29-May-2021.)
Hypotheses
Ref Expression
ntrnei.o 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
ntrnei.f 𝐹 = (𝒫 𝐵𝑂𝐵)
ntrnei.r (𝜑𝐼𝐹𝑁)
Assertion
Ref Expression
ntrneifv2 (𝜑 → (𝐹𝑁) = 𝐼)
Distinct variable groups:   𝐵,𝑖,𝑗,𝑘,𝑙,𝑚   𝜑,𝑖,𝑗,𝑘,𝑙
Allowed substitution hints:   𝜑(𝑚)   𝐹(𝑖,𝑗,𝑘,𝑚,𝑙)   𝐼(𝑖,𝑗,𝑘,𝑚,𝑙)   𝑁(𝑖,𝑗,𝑘,𝑚,𝑙)   𝑂(𝑖,𝑗,𝑘,𝑚,𝑙)

Proof of Theorem ntrneifv2
StepHypRef Expression
1 ntrnei.r . 2 (𝜑𝐼𝐹𝑁)
2 ntrnei.o . . . . . 6 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
3 ntrnei.f . . . . . 6 𝐹 = (𝒫 𝐵𝑂𝐵)
42, 3, 1ntrneif1o 37852 . . . . 5 (𝜑𝐹:(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵𝑚 𝐵))
52, 3, 1ntrneinex 37854 . . . . 5 (𝜑𝑁 ∈ (𝒫 𝒫 𝐵𝑚 𝐵))
6 dff1o3 6100 . . . . . . . 8 (𝐹:(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵𝑚 𝐵) ↔ (𝐹:(𝒫 𝐵𝑚 𝒫 𝐵)–onto→(𝒫 𝒫 𝐵𝑚 𝐵) ∧ Fun 𝐹))
76simprbi 480 . . . . . . 7 (𝐹:(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵𝑚 𝐵) → Fun 𝐹)
87adantr 481 . . . . . 6 ((𝐹:(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵𝑚 𝐵) ∧ 𝑁 ∈ (𝒫 𝒫 𝐵𝑚 𝐵)) → Fun 𝐹)
9 df-rn 5085 . . . . . . . . 9 ran 𝐹 = dom 𝐹
10 f1ofo 6101 . . . . . . . . . 10 (𝐹:(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵𝑚 𝐵) → 𝐹:(𝒫 𝐵𝑚 𝒫 𝐵)–onto→(𝒫 𝒫 𝐵𝑚 𝐵))
11 forn 6075 . . . . . . . . . 10 (𝐹:(𝒫 𝐵𝑚 𝒫 𝐵)–onto→(𝒫 𝒫 𝐵𝑚 𝐵) → ran 𝐹 = (𝒫 𝒫 𝐵𝑚 𝐵))
1210, 11syl 17 . . . . . . . . 9 (𝐹:(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵𝑚 𝐵) → ran 𝐹 = (𝒫 𝒫 𝐵𝑚 𝐵))
139, 12syl5eqr 2669 . . . . . . . 8 (𝐹:(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵𝑚 𝐵) → dom 𝐹 = (𝒫 𝒫 𝐵𝑚 𝐵))
1413eleq2d 2684 . . . . . . 7 (𝐹:(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵𝑚 𝐵) → (𝑁 ∈ dom 𝐹𝑁 ∈ (𝒫 𝒫 𝐵𝑚 𝐵)))
1514biimpar 502 . . . . . 6 ((𝐹:(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵𝑚 𝐵) ∧ 𝑁 ∈ (𝒫 𝒫 𝐵𝑚 𝐵)) → 𝑁 ∈ dom 𝐹)
168, 15jca 554 . . . . 5 ((𝐹:(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵𝑚 𝐵) ∧ 𝑁 ∈ (𝒫 𝒫 𝐵𝑚 𝐵)) → (Fun 𝐹𝑁 ∈ dom 𝐹))
174, 5, 16syl2anc 692 . . . 4 (𝜑 → (Fun 𝐹𝑁 ∈ dom 𝐹))
18 funbrfvb 6195 . . . 4 ((Fun 𝐹𝑁 ∈ dom 𝐹) → ((𝐹𝑁) = 𝐼𝑁𝐹𝐼))
1917, 18syl 17 . . 3 (𝜑 → ((𝐹𝑁) = 𝐼𝑁𝐹𝐼))
202, 3, 1ntrneiiex 37853 . . . 4 (𝜑𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
21 brcnvg 5263 . . . 4 ((𝑁 ∈ (𝒫 𝒫 𝐵𝑚 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → (𝑁𝐹𝐼𝐼𝐹𝑁))
225, 20, 21syl2anc 692 . . 3 (𝜑 → (𝑁𝐹𝐼𝐼𝐹𝑁))
2319, 22bitrd 268 . 2 (𝜑 → ((𝐹𝑁) = 𝐼𝐼𝐹𝑁))
241, 23mpbird 247 1 (𝜑 → (𝐹𝑁) = 𝐼)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  {crab 2911  Vcvv 3186  𝒫 cpw 4130   class class class wbr 4613  cmpt 4673  ccnv 5073  dom cdm 5074  ran crn 5075  Fun wfun 5841  ontowfo 5845  1-1-ontowf1o 5846  cfv 5847  (class class class)co 6604  cmpt2 6606  𝑚 cmap 7802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-map 7804
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator