Theorem ntrneifv1 39914

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 39910	. . . . 5 ⊢ (𝜑 → 𝐹:(𝒫 𝐵 ↑𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑𝑚 𝐵))
5		f1ofn 6484	. . . . 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 39911	. . . 4 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵))
8	6, 7	jca 512	. . 3 ⊢ (𝜑 → (𝐹 Fn (𝒫 𝐵 ↑𝑚 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵)))
9		fnfun 6323	. . . . 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 6325	. . . . . 6 ⊢ (𝐹 Fn (𝒫 𝐵 ↑𝑚 𝒫 𝐵) → dom 𝐹 = (𝒫 𝐵 ↑𝑚 𝒫 𝐵))
12	11	eleq2d 2868	. . . . 5 ⊢ (𝐹 Fn (𝒫 𝐵 ↑𝑚 𝒫 𝐵) → (𝐼 ∈ dom 𝐹 ↔ 𝐼 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵)))
13	12	biimpar 478	. . . 4 ⊢ ((𝐹 Fn (𝒫 𝐵 ↑𝑚 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵)) → 𝐼 ∈ dom 𝐹)
14	10, 13	jca 512	. . 3 ⊢ ((𝐹 Fn (𝒫 𝐵 ↑𝑚 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵)) → (Fun 𝐹 ∧ 𝐼 ∈ dom 𝐹))
15		funbrfvb 6588	. . 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 258	1 ⊢ (𝜑 → (𝐹‘𝐼) = 𝑁)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1522   ∈ wcel 2081  {crab 3109  Vcvv 3437  𝒫 cpw 4453   class class class wbr 4962   ↦ cmpt 5041  dom cdm 5443  Fun wfun 6219   Fn wfn 6220  –1-1-onto→wf1o 6224  ‘cfv 6225  (class class class)co 7016   ∈ cmpo 7018   ↑_𝑚 cmap 8256

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-ov 7019  df-oprab 7020  df-mpo 7021  df-1st 7545  df-2nd 7546  df-map 8258

This theorem is referenced by:  ntrneiel  39916