Theorem ntrf 45122

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 5339	. . . . . 6 ⊢ 𝒫 𝑠 ∈ V
2	1	inex2 5278	. . . . 5 ⊢ (𝐽 ∩ 𝒫 𝑠) ∈ V
3	2	uniex 7758	. . . 4 ⊢ ∪ (𝐽 ∩ 𝒫 𝑠) ∈ V
4		eqid 2761	. . . 4 ⊢ (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)) = (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠))
5	3, 4	fnmpti 6682	. . 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 23342	. . . . 5 ⊢ (𝐽 ∈ Top → (int‘𝐽) = (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)))
9	6, 8	eqtrid 2808	. . . 4 ⊢ (𝐽 ∈ Top → 𝐼 = (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)))
10	9	fneq1d 6632	. . 3 ⊢ (𝐽 ∈ Top → (𝐼 Fn 𝒫 𝑋 ↔ (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)) Fn 𝒫 𝑋))
11	5, 10	mpbiri 261	. 2 ⊢ (𝐽 ∈ Top → 𝐼 Fn 𝒫 𝑋)
12	7, 6	ntrrn 45121	. 2 ⊢ (𝐽 ∈ Top → ran 𝐼 ⊆ 𝐽)
13		df-f 6542	. 2 ⊢ (𝐼:𝒫 𝑋⟶𝐽 ↔ (𝐼 Fn 𝒫 𝑋 ∧ ran 𝐼 ⊆ 𝐽))
14	11, 12, 13	sylanbrc 595	1 ⊢ (𝐽 ∈ Top → 𝐼:𝒫 𝑋⟶𝐽)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145   ∩ cin 3898   ⊆ wss 3899  𝒫 cpw 4557  ∪ cuni 4867   ↦ cmpt 5186  ran crn 5652   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  Topctop 23211  intcnt 23335

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-top 23212  df-ntr 23338

This theorem is used by:  ntrf2  45123