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Theorem ntrneifv2 38198
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 38193 . . . . 5 (𝜑𝐹:(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵𝑚 𝐵))
52, 3, 1ntrneinex 38195 . . . . 5 (𝜑𝑁 ∈ (𝒫 𝒫 𝐵𝑚 𝐵))
6 dff1o3 6130 . . . . . . . 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 5115 . . . . . . . . 9 ran 𝐹 = dom 𝐹
10 f1ofo 6131 . . . . . . . . . 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 6105 . . . . . . . . . 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 2668 . . . . . . . 8 (𝐹:(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵𝑚 𝐵) → dom 𝐹 = (𝒫 𝒫 𝐵𝑚 𝐵))
1413eleq2d 2685 . . . . . . 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 6225 . . . 4 ((Fun 𝐹𝑁 ∈ dom 𝐹) → ((𝐹𝑁) = 𝐼𝑁𝐹𝐼))
1917, 18syl 17 . . 3 (𝜑 → ((𝐹𝑁) = 𝐼𝑁𝐹𝐼))
202, 3, 1ntrneiiex 38194 . . . 4 (𝜑𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
21 brcnvg 5292 . . . 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 1481  wcel 1988  {crab 2913  Vcvv 3195  𝒫 cpw 4149   class class class wbr 4644  cmpt 4720  ccnv 5103  dom cdm 5104  ran crn 5105  Fun wfun 5870  ontowfo 5874  1-1-ontowf1o 5875  cfv 5876  (class class class)co 6635  cmpt2 6637  𝑚 cmap 7842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-1st 7153  df-2nd 7154  df-map 7844
This theorem is referenced by: (None)
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