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Theorem ntrneifv2 40567
 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 40562 . . . . 5 (𝜑𝐹:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵m 𝐵))
52, 3, 1ntrneinex 40564 . . . . 5 (𝜑𝑁 ∈ (𝒫 𝒫 𝐵m 𝐵))
6 dff1o3 6594 . . . . . . . 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 500 . . . . . . 7 (𝐹:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵m 𝐵) → Fun 𝐹)
87adantr 484 . . . . . 6 ((𝐹:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵m 𝐵) ∧ 𝑁 ∈ (𝒫 𝒫 𝐵m 𝐵)) → Fun 𝐹)
9 df-rn 5539 . . . . . . . . 9 ran 𝐹 = dom 𝐹
10 f1ofo 6595 . . . . . . . . . 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 6566 . . . . . . . . . 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 2870 . . . . . . . 8 (𝐹:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵m 𝐵) → dom 𝐹 = (𝒫 𝒫 𝐵m 𝐵))
1413eleq2d 2897 . . . . . . 7 (𝐹:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵m 𝐵) → (𝑁 ∈ dom 𝐹𝑁 ∈ (𝒫 𝒫 𝐵m 𝐵)))
1514biimpar 481 . . . . . 6 ((𝐹:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵m 𝐵) ∧ 𝑁 ∈ (𝒫 𝒫 𝐵m 𝐵)) → 𝑁 ∈ dom 𝐹)
168, 15jca 515 . . . . 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 587 . . . 4 (𝜑 → (Fun 𝐹𝑁 ∈ dom 𝐹))
18 funbrfvb 6693 . . . 4 ((Fun 𝐹𝑁 ∈ dom 𝐹) → ((𝐹𝑁) = 𝐼𝑁𝐹𝐼))
1917, 18syl 17 . . 3 (𝜑 → ((𝐹𝑁) = 𝐼𝑁𝐹𝐼))
202, 3, 1ntrneiiex 40563 . . . 4 (𝜑𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
21 brcnvg 5723 . . . 4 ((𝑁 ∈ (𝒫 𝒫 𝐵m 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵)) → (𝑁𝐹𝐼𝐼𝐹𝑁))
225, 20, 21syl2anc 587 . . 3 (𝜑 → (𝑁𝐹𝐼𝐼𝐹𝑁))
2319, 22bitrd 282 . 2 (𝜑 → ((𝐹𝑁) = 𝐼𝐼𝐹𝑁))
241, 23mpbird 260 1 (𝜑 → (𝐹𝑁) = 𝐼)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2115  {crab 3130  Vcvv 3471  𝒫 cpw 4512   class class class wbr 5039   ↦ cmpt 5119  ◡ccnv 5527  dom cdm 5528  ran crn 5529  Fun wfun 6322  –onto→wfo 6326  –1-1-onto→wf1o 6327  ‘cfv 6328  (class class class)co 7130   ∈ cmpo 7132   ↑m cmap 8381 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-1st 7664  df-2nd 7665  df-map 8383 This theorem is referenced by: (None)
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