Theorem ntrneifv4 40442

Description: The value of the interior (closure) expressed in terms of the neighbors (convergents) function. (Contributed by RP, 26-Jun-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)
ntrneifv.s	⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)

Assertion

Ref	Expression
ntrneifv4	⊢ (𝜑 → (𝐼‘𝑆) = {𝑥 ∈ 𝐵 ∣ 𝑆 ∈ (𝑁‘𝑥)})

Distinct variable groups:   𝐵,𝑖,𝑗,𝑘,𝑙,𝑚,𝑥   𝑘,𝐼,𝑙,𝑚,𝑥   𝑆,𝑚,𝑥   𝜑,𝑖,𝑗,𝑘,𝑙,𝑥

Allowed substitution hints:   𝜑(𝑚)   𝑆(𝑖,𝑗,𝑘,𝑙)   𝐹(𝑥,𝑖,𝑗,𝑘,𝑚,𝑙)   𝐼(𝑖,𝑗)   𝑁(𝑥,𝑖,𝑗,𝑘,𝑚,𝑙)   𝑂(𝑥,𝑖,𝑗,𝑘,𝑚,𝑙)

Proof of Theorem ntrneifv4

Step	Hyp	Ref	Expression
1		dfin5 3946	. 2 ⊢ (𝐵 ∩ (𝐼‘𝑆)) = {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ (𝐼‘𝑆)}
2		ntrnei.o	. . . . . . 7 ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))
3		ntrnei.f	. . . . . . 7 ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)
4		ntrnei.r	. . . . . . 7 ⊢ (𝜑 → 𝐼𝐹𝑁)
5	2, 3, 4	ntrneiiex 40433	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
6		elmapi 8430	. . . . . 6 ⊢ (𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
7	5, 6	syl 17	. . . . 5 ⊢ (𝜑 → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
8		ntrneifv.s	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)
9	7, 8	ffvelrnd 6854	. . . 4 ⊢ (𝜑 → (𝐼‘𝑆) ∈ 𝒫 𝐵)
10	9	elpwid 4552	. . 3 ⊢ (𝜑 → (𝐼‘𝑆) ⊆ 𝐵)
11		sseqin2 4194	. . 3 ⊢ ((𝐼‘𝑆) ⊆ 𝐵 ↔ (𝐵 ∩ (𝐼‘𝑆)) = (𝐼‘𝑆))
12	10, 11	sylib 220	. 2 ⊢ (𝜑 → (𝐵 ∩ (𝐼‘𝑆)) = (𝐼‘𝑆))
13	4	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐼𝐹𝑁)
14		simpr 487	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
15	8	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑆 ∈ 𝒫 𝐵)
16	2, 3, 13, 14, 15	ntrneiel 40438	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ (𝐼‘𝑆) ↔ 𝑆 ∈ (𝑁‘𝑥)))
17	16	rabbidva 3480	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ (𝐼‘𝑆)} = {𝑥 ∈ 𝐵 ∣ 𝑆 ∈ (𝑁‘𝑥)})
18	1, 12, 17	3eqtr3a 2882	1 ⊢ (𝜑 → (𝐼‘𝑆) = {𝑥 ∈ 𝐵 ∣ 𝑆 ∈ (𝑁‘𝑥)})

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  {crab 3144  Vcvv 3496   ∩ cin 3937   ⊆ wss 3938  𝒫 cpw 4541   class class class wbr 5068   ↦ cmpt 5148  ⟶wf 6353  ‘cfv 6357  (class class class)co 7158   ∈ cmpo 7160   ↑_m cmap 8408

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-1st 7691  df-2nd 7692  df-map 8410

This theorem is referenced by:  ntrneiel2  40443