Theorem ntrneifv2 45220

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 45215	. . . . 5 ⊢ (𝜑 → 𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵))
5	2, 3, 1	ntrneinex 45217	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵))
6		dff1o3 6839	. . . . . . . 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 503	. . . . . . 7 ⊢ (𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵) → Fun ◡𝐹)
8	7	adantr 486	. . . . . 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 5665	. . . . . . . . 9 ⊢ ran 𝐹 = dom ◡𝐹
10		f1ofo 6840	. . . . . . . . . 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 6807	. . . . . . . . . 10 ⊢ (𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–onto→(𝒫 𝒫 𝐵 ↑m 𝐵) → ran 𝐹 = (𝒫 𝒫 𝐵 ↑m 𝐵))
12	10, 11	syl 18	. . . . . . . . 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 2812	. . . . . . . 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 2849	. . . . . . 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 483	. . . . . 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 521	. . . . 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 596	. . . 4 ⊢ (𝜑 → (Fun ◡𝐹 ∧ 𝑁 ∈ dom ◡𝐹))
18		funbrfvb 6946	. . . 4 ⊢ ((Fun ◡𝐹 ∧ 𝑁 ∈ dom ◡𝐹) → ((◡𝐹‘𝑁) = 𝐼 ↔ 𝑁◡𝐹𝐼))
19	17, 18	syl 18	. . 3 ⊢ (𝜑 → ((◡𝐹‘𝑁) = 𝐼 ↔ 𝑁◡𝐹𝐼))
20	2, 3, 1	ntrneiiex 45216	. . . 4 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
21		brcnvg 5861	. . . 4 ⊢ ((𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → (𝑁◡𝐹𝐼 ↔ 𝐼𝐹𝑁))
22	5, 20, 21	syl2anc 596	. . 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)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2146  {crab 3415  Vcvv 3453  𝒫 cpw 4559   class class class wbr 5105   ↦ cmpt 5188  ^◡ccnv 5653  dom cdm 5654  ran crn 5655  Fun wfun 6540  –onto→wfo 6544  –1-1-onto→wf1o 6545  ‘cfv 6546  (class class class)co 7427   ∈ cmpo 7429   ↑_m cmap 8858

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5234  ax-sep 5251  ax-nul 5262  ax-pow 5330  ax-pr 5394  ax-un 7758

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3369  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-iun 4955  df-br 5106  df-opab 5170  df-mpt 5189  df-id 5549  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-ov 7430  df-oprab 7431  df-mpo 7432  df-1st 8008  df-2nd 8009  df-map 8860

This theorem is used by: (None)