Theorem ntrneifv2 40783

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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))
ntrnei.f	⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)
ntrnei.r	⊢ (𝜑 → 𝐼𝐹𝑁)

Assertion

Ref	Expression
ntrneifv2	⊢ (𝜑 → (◡𝐹‘𝑁) = 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

Step	Hyp	Ref	Expression
1		ntrnei.r	. 2 ⊢ (𝜑 → 𝐼𝐹𝑁)
2		ntrnei.o	. . . . . 6 ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))
3		ntrnei.f	. . . . . 6 ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)
4	2, 3, 1	ntrneif1o 40778	. . . . 5 ⊢ (𝜑 → 𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵))
5	2, 3, 1	ntrneinex 40780	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵))
6		dff1o3 6596	. . . . . . . 8 ⊢ (𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵) ↔ (𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–onto→(𝒫 𝒫 𝐵 ↑m 𝐵) ∧ Fun ◡𝐹))
7	6	simprbi 500	. . . . . . 7 ⊢ (𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵) → Fun ◡𝐹)
8	7	adantr 484	. . . . . 6 ⊢ ((𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵) ∧ 𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵)) → Fun ◡𝐹)
9		df-rn 5530	. . . . . . . . 9 ⊢ ran 𝐹 = dom ◡𝐹
10		f1ofo 6597	. . . . . . . . . 10 ⊢ (𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵) → 𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–onto→(𝒫 𝒫 𝐵 ↑m 𝐵))
11		forn 6568	. . . . . . . . . 10 ⊢ (𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–onto→(𝒫 𝒫 𝐵 ↑m 𝐵) → ran 𝐹 = (𝒫 𝒫 𝐵 ↑m 𝐵))
12	10, 11	syl 17	. . . . . . . . 9 ⊢ (𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵) → ran 𝐹 = (𝒫 𝒫 𝐵 ↑m 𝐵))
13	9, 12	syl5eqr 2847	. . . . . . . 8 ⊢ (𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵) → dom ◡𝐹 = (𝒫 𝒫 𝐵 ↑m 𝐵))
14	13	eleq2d 2875	. . . . . . 7 ⊢ (𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵) → (𝑁 ∈ dom ◡𝐹 ↔ 𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵)))
15	14	biimpar 481	. . . . . 6 ⊢ ((𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵) ∧ 𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵)) → 𝑁 ∈ dom ◡𝐹)
16	8, 15	jca 515	. . . . 5 ⊢ ((𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵) ∧ 𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵)) → (Fun ◡𝐹 ∧ 𝑁 ∈ dom ◡𝐹))
17	4, 5, 16	syl2anc 587	. . . 4 ⊢ (𝜑 → (Fun ◡𝐹 ∧ 𝑁 ∈ dom ◡𝐹))
18		funbrfvb 6695	. . . 4 ⊢ ((Fun ◡𝐹 ∧ 𝑁 ∈ dom ◡𝐹) → ((◡𝐹‘𝑁) = 𝐼 ↔ 𝑁◡𝐹𝐼))
19	17, 18	syl 17	. . 3 ⊢ (𝜑 → ((◡𝐹‘𝑁) = 𝐼 ↔ 𝑁◡𝐹𝐼))
20	2, 3, 1	ntrneiiex 40779	. . . 4 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
21		brcnvg 5714	. . . 4 ⊢ ((𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → (𝑁◡𝐹𝐼 ↔ 𝐼𝐹𝑁))
22	5, 20, 21	syl2anc 587	. . 3 ⊢ (𝜑 → (𝑁◡𝐹𝐼 ↔ 𝐼𝐹𝑁))
23	19, 22	bitrd 282	. 2 ⊢ (𝜑 → ((◡𝐹‘𝑁) = 𝐼 ↔ 𝐼𝐹𝑁))
24	1, 23	mpbird 260	1 ⊢ (𝜑 → (◡𝐹‘𝑁) = 𝐼)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {crab 3110  Vcvv 3441  𝒫 cpw 4497   class class class wbr 5030   ↦ cmpt 5110  ^◡ccnv 5518  dom cdm 5519  ran crn 5520  Fun wfun 6318  –onto→wfo 6322  –1-1-onto→wf1o 6323  ‘cfv 6324  (class class class)co 7135   ∈ cmpo 7137   ↑_m cmap 8389

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7671  df-2nd 7672  df-map 8391

This theorem is referenced by: (None)