Theorem ntrf 43695

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 5377	. . . . . 6 ⊢ 𝒫 𝑠 ∈ V
2	1	inex2 5319	. . . . 5 ⊢ (𝐽 ∩ 𝒫 𝑠) ∈ V
3	2	uniex 7747	. . . 4 ⊢ ∪ (𝐽 ∩ 𝒫 𝑠) ∈ V
4		eqid 2725	. . . 4 ⊢ (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)) = (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠))
5	3, 4	fnmpti 6699	. . 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 22972	. . . . 5 ⊢ (𝐽 ∈ Top → (int‘𝐽) = (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)))
9	6, 8	eqtrid 2777	. . . 4 ⊢ (𝐽 ∈ Top → 𝐼 = (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)))
10	9	fneq1d 6648	. . 3 ⊢ (𝐽 ∈ Top → (𝐼 Fn 𝒫 𝑋 ↔ (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)) Fn 𝒫 𝑋))
11	5, 10	mpbiri 257	. 2 ⊢ (𝐽 ∈ Top → 𝐼 Fn 𝒫 𝑋)
12	7, 6	ntrrn 43694	. 2 ⊢ (𝐽 ∈ Top → ran 𝐼 ⊆ 𝐽)
13		df-f 6553	. 2 ⊢ (𝐼:𝒫 𝑋⟶𝐽 ↔ (𝐼 Fn 𝒫 𝑋 ∧ ran 𝐼 ⊆ 𝐽))
14	11, 12, 13	sylanbrc 581	1 ⊢ (𝐽 ∈ Top → 𝐼:𝒫 𝑋⟶𝐽)

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098   ∩ cin 3943   ⊆ wss 3944  𝒫 cpw 4604  ∪ cuni 4909   ↦ cmpt 5232  ran crn 5679   Fn wfn 6544  ⟶wf 6545  ‘cfv 6549  Topctop 22839  intcnt 22965

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-top 22840  df-ntr 22968

This theorem is referenced by:  ntrf2  43696