Theorem ntrf 44085

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 5327	. . . . . 6 ⊢ 𝒫 𝑠 ∈ V
2	1	inex2 5268	. . . . 5 ⊢ (𝐽 ∩ 𝒫 𝑠) ∈ V
3	2	uniex 7697	. . . 4 ⊢ ∪ (𝐽 ∩ 𝒫 𝑠) ∈ V
4		eqid 2729	. . . 4 ⊢ (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)) = (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠))
5	3, 4	fnmpti 6643	. . 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 22887	. . . . 5 ⊢ (𝐽 ∈ Top → (int‘𝐽) = (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)))
9	6, 8	eqtrid 2776	. . . 4 ⊢ (𝐽 ∈ Top → 𝐼 = (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)))
10	9	fneq1d 6593	. . 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 44084	. 2 ⊢ (𝐽 ∈ Top → ran 𝐼 ⊆ 𝐽)
13		df-f 6503	. 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 1540   ∈ wcel 2109   ∩ cin 3910   ⊆ wss 3911  𝒫 cpw 4559  ∪ cuni 4867   ↦ cmpt 5183  ran crn 5632   Fn wfn 6494  ⟶wf 6495  ‘cfv 6499  Topctop 22756  intcnt 22880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-top 22757  df-ntr 22883

This theorem is referenced by:  ntrf2  44086