Theorem ntrf 44113

Description: The interior function of a topology is a map from the powerset of the base set to the open sets of the topology. (Contributed by RP, 22-Apr-2021.)

Hypotheses

Ref	Expression
ntrrn.x	⊢ 𝑋 = ∪ 𝐽
ntrrn.i	⊢ 𝐼 = (int‘𝐽)

Assertion

Ref	Expression
ntrf	⊢ (𝐽 ∈ Top → 𝐼:𝒫 𝑋⟶𝐽)

Proof of Theorem ntrf

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vpwex 5383	. . . . . 6 ⊢ 𝒫 𝑠 ∈ V
2	1	inex2 5324	. . . . 5 ⊢ (𝐽 ∩ 𝒫 𝑠) ∈ V
3	2	uniex 7760	. . . 4 ⊢ ∪ (𝐽 ∩ 𝒫 𝑠) ∈ V
4		eqid 2735	. . . 4 ⊢ (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)) = (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠))
5	3, 4	fnmpti 6712	. . 3 ⊢ (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)) Fn 𝒫 𝑋
6		ntrrn.i	. . . . 5 ⊢ 𝐼 = (int‘𝐽)
7		ntrrn.x	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
8	7	ntrfval 23048	. . . . 5 ⊢ (𝐽 ∈ Top → (int‘𝐽) = (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)))
9	6, 8	eqtrid 2787	. . . 4 ⊢ (𝐽 ∈ Top → 𝐼 = (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)))
10	9	fneq1d 6662	. . 3 ⊢ (𝐽 ∈ Top → (𝐼 Fn 𝒫 𝑋 ↔ (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)) Fn 𝒫 𝑋))
11	5, 10	mpbiri 258	. 2 ⊢ (𝐽 ∈ Top → 𝐼 Fn 𝒫 𝑋)
12	7, 6	ntrrn 44112	. 2 ⊢ (𝐽 ∈ Top → ran 𝐼 ⊆ 𝐽)
13		df-f 6567	. 2 ⊢ (𝐼:𝒫 𝑋⟶𝐽 ↔ (𝐼 Fn 𝒫 𝑋 ∧ ran 𝐼 ⊆ 𝐽))
14	11, 12, 13	sylanbrc 583	1 ⊢ (𝐽 ∈ Top → 𝐼:𝒫 𝑋⟶𝐽)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106   ∩ cin 3962   ⊆ wss 3963  𝒫 cpw 4605  ∪ cuni 4912   ↦ cmpt 5231  ran crn 5690   Fn wfn 6558  ⟶wf 6559  ‘cfv 6563  Topctop 22915  intcnt 23041

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-top 22916  df-ntr 23044

This theorem is referenced by:  ntrf2  44114