Theorem ntrf 44567

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 5306	. . . . . 6 ⊢ 𝒫 𝑠 ∈ V
2	1	inex2 5246	. . . . 5 ⊢ (𝐽 ∩ 𝒫 𝑠) ∈ V
3	2	uniex 7684	. . . 4 ⊢ ∪ (𝐽 ∩ 𝒫 𝑠) ∈ V
4		eqid 2739	. . . 4 ⊢ (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)) = (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠))
5	3, 4	fnmpti 6628	. . 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 23007	. . . . 5 ⊢ (𝐽 ∈ Top → (int‘𝐽) = (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)))
9	6, 8	eqtrid 2786	. . . 4 ⊢ (𝐽 ∈ Top → 𝐼 = (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)))
10	9	fneq1d 6578	. . 3 ⊢ (𝐽 ∈ Top → (𝐼 Fn 𝒫 𝑋 ↔ (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)) Fn 𝒫 𝑋))
11	5, 10	mpbiri 259	. 2 ⊢ (𝐽 ∈ Top → 𝐼 Fn 𝒫 𝑋)
12	7, 6	ntrrn 44566	. 2 ⊢ (𝐽 ∈ Top → ran 𝐼 ⊆ 𝐽)
13		df-f 6489	. 2 ⊢ (𝐼:𝒫 𝑋⟶𝐽 ↔ (𝐼 Fn 𝒫 𝑋 ∧ ran 𝐼 ⊆ 𝐽))
14	11, 12, 13	sylanbrc 589	1 ⊢ (𝐽 ∈ Top → 𝐼:𝒫 𝑋⟶𝐽)

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119   ∩ cin 3882   ⊆ wss 3883  𝒫 cpw 4529  ∪ cuni 4838   ↦ cmpt 5153  ran crn 5619   Fn wfn 6480  ⟶wf 6481  ‘cfv 6485  Topctop 22876  intcnt 23000

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-top 22877  df-ntr 23003

This theorem is referenced by:  ntrf2  44568