Theorem ntrneifv2 44183

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 44178	. . . . 5 ⊢ (𝜑 → 𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵))
5	2, 3, 1	ntrneinex 44180	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵))
6		dff1o3 6769	. . . . . . . 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 496	. . . . . . 7 ⊢ (𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵) → Fun ◡𝐹)
8	7	adantr 480	. . . . . 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 5625	. . . . . . . . 9 ⊢ ran 𝐹 = dom ◡𝐹
10		f1ofo 6770	. . . . . . . . . 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 6738	. . . . . . . . . 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	eqtr3id 2780	. . . . . . . 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 2817	. . . . . . 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 477	. . . . . 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 511	. . . . 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 584	. . . 4 ⊢ (𝜑 → (Fun ◡𝐹 ∧ 𝑁 ∈ dom ◡𝐹))
18		funbrfvb 6875	. . . 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 44179	. . . 4 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
21		brcnvg 5818	. . . 4 ⊢ ((𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → (𝑁◡𝐹𝐼 ↔ 𝐼𝐹𝑁))
22	5, 20, 21	syl2anc 584	. . 3 ⊢ (𝜑 → (𝑁◡𝐹𝐼 ↔ 𝐼𝐹𝑁))
23	19, 22	bitrd 279	. 2 ⊢ (𝜑 → ((◡𝐹‘𝑁) = 𝐼 ↔ 𝐼𝐹𝑁))
24	1, 23	mpbird 257	1 ⊢ (𝜑 → (◡𝐹‘𝑁) = 𝐼)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {crab 3395  Vcvv 3436  𝒫 cpw 4547   class class class wbr 5089   ↦ cmpt 5170  ^◡ccnv 5613  dom cdm 5614  ran crn 5615  Fun wfun 6475  –onto→wfo 6479  –1-1-onto→wf1o 6480  ‘cfv 6481  (class class class)co 7346   ∈ cmpo 7348   ↑_m cmap 8750

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-map 8752

This theorem is referenced by: (None)