Theorem ntrneifv1 38377

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 38373	. . . . 5 ⊢ (𝜑 → 𝐹:(𝒫 𝐵 ↑𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑𝑚 𝐵))
5		f1ofn 6138	. . . . 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 17	. . . 4 ⊢ (𝜑 → 𝐹 Fn (𝒫 𝐵 ↑𝑚 𝒫 𝐵))
7	2, 3, 1	ntrneiiex 38374	. . . 4 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵))
8	6, 7	jca 554	. . 3 ⊢ (𝜑 → (𝐹 Fn (𝒫 𝐵 ↑𝑚 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵)))
9		fnfun 5988	. . . . 5 ⊢ (𝐹 Fn (𝒫 𝐵 ↑𝑚 𝒫 𝐵) → Fun 𝐹)
10	9	adantr 481	. . . 4 ⊢ ((𝐹 Fn (𝒫 𝐵 ↑𝑚 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵)) → Fun 𝐹)
11		fndm 5990	. . . . . 6 ⊢ (𝐹 Fn (𝒫 𝐵 ↑𝑚 𝒫 𝐵) → dom 𝐹 = (𝒫 𝐵 ↑𝑚 𝒫 𝐵))
12	11	eleq2d 2687	. . . . 5 ⊢ (𝐹 Fn (𝒫 𝐵 ↑𝑚 𝒫 𝐵) → (𝐼 ∈ dom 𝐹 ↔ 𝐼 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵)))
13	12	biimpar 502	. . . 4 ⊢ ((𝐹 Fn (𝒫 𝐵 ↑𝑚 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵)) → 𝐼 ∈ dom 𝐹)
14	10, 13	jca 554	. . 3 ⊢ ((𝐹 Fn (𝒫 𝐵 ↑𝑚 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵)) → (Fun 𝐹 ∧ 𝐼 ∈ dom 𝐹))
15		funbrfvb 6238	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐼 ∈ dom 𝐹) → ((𝐹‘𝐼) = 𝑁 ↔ 𝐼𝐹𝑁))
16	8, 14, 15	3syl 18	. 2 ⊢ (𝜑 → ((𝐹‘𝐼) = 𝑁 ↔ 𝐼𝐹𝑁))
17	1, 16	mpbird 247	1 ⊢ (𝜑 → (𝐹‘𝐼) = 𝑁)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {crab 2916  Vcvv 3200  𝒫 cpw 4158   class class class wbr 4653   ↦ cmpt 4729  dom cdm 5114  Fun wfun 5882   Fn wfn 5883  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652   ↑_𝑚 cmap 7857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859

This theorem is referenced by:  ntrneiel  38379