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Theorem ntrneifv2 42831
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 42826 . . . . 5 (𝜑𝐹:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵m 𝐵))
52, 3, 1ntrneinex 42828 . . . . 5 (𝜑𝑁 ∈ (𝒫 𝒫 𝐵m 𝐵))
6 dff1o3 6840 . . . . . . . 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 498 . . . . . . 7 (𝐹:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵m 𝐵) → Fun 𝐹)
87adantr 482 . . . . . 6 ((𝐹:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵m 𝐵) ∧ 𝑁 ∈ (𝒫 𝒫 𝐵m 𝐵)) → Fun 𝐹)
9 df-rn 5688 . . . . . . . . 9 ran 𝐹 = dom 𝐹
10 f1ofo 6841 . . . . . . . . . 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 6809 . . . . . . . . . 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 2787 . . . . . . . 8 (𝐹:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵m 𝐵) → dom 𝐹 = (𝒫 𝒫 𝐵m 𝐵))
1413eleq2d 2820 . . . . . . 7 (𝐹:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵m 𝐵) → (𝑁 ∈ dom 𝐹𝑁 ∈ (𝒫 𝒫 𝐵m 𝐵)))
1514biimpar 479 . . . . . 6 ((𝐹:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵m 𝐵) ∧ 𝑁 ∈ (𝒫 𝒫 𝐵m 𝐵)) → 𝑁 ∈ dom 𝐹)
168, 15jca 513 . . . . 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 585 . . . 4 (𝜑 → (Fun 𝐹𝑁 ∈ dom 𝐹))
18 funbrfvb 6947 . . . 4 ((Fun 𝐹𝑁 ∈ dom 𝐹) → ((𝐹𝑁) = 𝐼𝑁𝐹𝐼))
1917, 18syl 17 . . 3 (𝜑 → ((𝐹𝑁) = 𝐼𝑁𝐹𝐼))
202, 3, 1ntrneiiex 42827 . . . 4 (𝜑𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
21 brcnvg 5880 . . . 4 ((𝑁 ∈ (𝒫 𝒫 𝐵m 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵)) → (𝑁𝐹𝐼𝐼𝐹𝑁))
225, 20, 21syl2anc 585 . . 3 (𝜑 → (𝑁𝐹𝐼𝐼𝐹𝑁))
2319, 22bitrd 279 . 2 (𝜑 → ((𝐹𝑁) = 𝐼𝐼𝐹𝑁))
241, 23mpbird 257 1 (𝜑 → (𝐹𝑁) = 𝐼)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  {crab 3433  Vcvv 3475  𝒫 cpw 4603   class class class wbr 5149  cmpt 5232  ccnv 5676  dom cdm 5677  ran crn 5678  Fun wfun 6538  ontowfo 6542  1-1-ontowf1o 6543  cfv 6544  (class class class)co 7409  cmpo 7411  m cmap 8820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-map 8822
This theorem is referenced by: (None)
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