Theorem ntrneifv1 44805

Description: If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then the function value of 𝐹 is the neighborhood 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
ntrneifv1	⊢ (𝜑 → (𝐹‘𝐼) = 𝑁)

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 ntrneifv1

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 44801	. . . . 5 ⊢ (𝜑 → 𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵))
5		f1ofn 6821	. . . . 5 ⊢ (𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵) → 𝐹 Fn (𝒫 𝐵 ↑m 𝒫 𝐵))
6	4, 5	syl 18	. . . 4 ⊢ (𝜑 → 𝐹 Fn (𝒫 𝐵 ↑m 𝒫 𝐵))
7	2, 3, 1	ntrneiiex 44802	. . . 4 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
8	6, 7	jca 520	. . 3 ⊢ (𝜑 → (𝐹 Fn (𝒫 𝐵 ↑m 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)))
9		fnfun 6635	. . . . 5 ⊢ (𝐹 Fn (𝒫 𝐵 ↑m 𝒫 𝐵) → Fun 𝐹)
10	9	adantr 485	. . . 4 ⊢ ((𝐹 Fn (𝒫 𝐵 ↑m 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → Fun 𝐹)
11		fndm 6638	. . . . . 6 ⊢ (𝐹 Fn (𝒫 𝐵 ↑m 𝒫 𝐵) → dom 𝐹 = (𝒫 𝐵 ↑m 𝒫 𝐵))
12	11	eleq2d 2849	. . . . 5 ⊢ (𝐹 Fn (𝒫 𝐵 ↑m 𝒫 𝐵) → (𝐼 ∈ dom 𝐹 ↔ 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)))
13	12	biimpar 482	. . . 4 ⊢ ((𝐹 Fn (𝒫 𝐵 ↑m 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → 𝐼 ∈ dom 𝐹)
14	10, 13	jca 520	. . 3 ⊢ ((𝐹 Fn (𝒫 𝐵 ↑m 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → (Fun 𝐹 ∧ 𝐼 ∈ dom 𝐹))
15		funbrfvb 6934	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐼 ∈ dom 𝐹) → ((𝐹‘𝐼) = 𝑁 ↔ 𝐼𝐹𝑁))
16	8, 14, 15	3syl 19	. 2 ⊢ (𝜑 → ((𝐹‘𝐼) = 𝑁 ↔ 𝐼𝐹𝑁))
17	1, 16	mpbird 260	1 ⊢ (𝜑 → (𝐹‘𝐼) = 𝑁)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  {crab 3416  Vcvv 3455  𝒫 cpw 4562   class class class wbr 5109   ↦ cmpt 5192  dom cdm 5661  Fun wfun 6530   Fn wfn 6531  –1-1-onto→wf1o 6535  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412   ↑_m cmap 8820

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  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 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-map 8822

This theorem is referenced by:  ntrneiel  44807