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Theorem ntrneifv2 43784
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 43779 . . . . 5 (𝜑𝐹:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵m 𝐵))
52, 3, 1ntrneinex 43781 . . . . 5 (𝜑𝑁 ∈ (𝒫 𝒫 𝐵m 𝐵))
6 dff1o3 6841 . . . . . . . 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 495 . . . . . . 7 (𝐹:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵m 𝐵) → Fun 𝐹)
87adantr 479 . . . . . 6 ((𝐹:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵m 𝐵) ∧ 𝑁 ∈ (𝒫 𝒫 𝐵m 𝐵)) → Fun 𝐹)
9 df-rn 5685 . . . . . . . . 9 ran 𝐹 = dom 𝐹
10 f1ofo 6842 . . . . . . . . . 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 6810 . . . . . . . . . 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, 12eqtr3id 2780 . . . . . . . 8 (𝐹:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵m 𝐵) → dom 𝐹 = (𝒫 𝒫 𝐵m 𝐵))
1413eleq2d 2812 . . . . . . 7 (𝐹:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵m 𝐵) → (𝑁 ∈ dom 𝐹𝑁 ∈ (𝒫 𝒫 𝐵m 𝐵)))
1514biimpar 476 . . . . . 6 ((𝐹:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵m 𝐵) ∧ 𝑁 ∈ (𝒫 𝒫 𝐵m 𝐵)) → 𝑁 ∈ dom 𝐹)
168, 15jca 510 . . . . 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 582 . . . 4 (𝜑 → (Fun 𝐹𝑁 ∈ dom 𝐹))
18 funbrfvb 6948 . . . 4 ((Fun 𝐹𝑁 ∈ dom 𝐹) → ((𝐹𝑁) = 𝐼𝑁𝐹𝐼))
1917, 18syl 17 . . 3 (𝜑 → ((𝐹𝑁) = 𝐼𝑁𝐹𝐼))
202, 3, 1ntrneiiex 43780 . . . 4 (𝜑𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
21 brcnvg 5878 . . . 4 ((𝑁 ∈ (𝒫 𝒫 𝐵m 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵)) → (𝑁𝐹𝐼𝐼𝐹𝑁))
225, 20, 21syl2anc 582 . . 3 (𝜑 → (𝑁𝐹𝐼𝐼𝐹𝑁))
2319, 22bitrd 278 . 2 (𝜑 → ((𝐹𝑁) = 𝐼𝐼𝐹𝑁))
241, 23mpbird 256 1 (𝜑 → (𝐹𝑁) = 𝐼)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1534  wcel 2099  {crab 3419  Vcvv 3462  𝒫 cpw 4597   class class class wbr 5145  cmpt 5228  ccnv 5673  dom cdm 5674  ran crn 5675  Fun wfun 6540  ontowfo 6544  1-1-ontowf1o 6545  cfv 6546  (class class class)co 7416  cmpo 7418  m cmap 8847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-iun 4995  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7995  df-2nd 7996  df-map 8849
This theorem is referenced by: (None)
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