Theorem ntrrn 44733

Description: The range of the interior function of a topology a subset of 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
ntrrn	⊢ (𝐽 ∈ Top → ran 𝐼 ⊆ 𝐽)

Proof of Theorem ntrrn

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ntrrn.i	. . 3 ⊢ 𝐼 = (int‘𝐽)
2	1	rneqi 5925	. 2 ⊢ ran 𝐼 = ran (int‘𝐽)
3		vpwex 5346	. . . . . . . 8 ⊢ 𝒫 𝑠 ∈ V
4	3	inex2 5286	. . . . . . 7 ⊢ (𝐽 ∩ 𝒫 𝑠) ∈ V
5	4	uniex 7736	. . . . . 6 ⊢ ∪ (𝐽 ∩ 𝒫 𝑠) ∈ V
6	5	rgenw 3089	. . . . 5 ⊢ ∀𝑠 ∈ 𝒫 𝑋∪ (𝐽 ∩ 𝒫 𝑠) ∈ V
7		nfcv 2931	. . . . . 6 ⊢ Ⅎ𝑠𝒫 𝑋
8	7	fnmptf 6669	. . . . 5 ⊢ (∀𝑠 ∈ 𝒫 𝑋∪ (𝐽 ∩ 𝒫 𝑠) ∈ V → (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)) Fn 𝒫 𝑋)
9	6, 8	mp1i 14	. . . 4 ⊢ (𝐽 ∈ Top → (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)) Fn 𝒫 𝑋)
10		ntrrn.x	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
11	10	ntrfval 23146	. . . . 5 ⊢ (𝐽 ∈ Top → (int‘𝐽) = (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)))
12	11	fneq1d 6626	. . . 4 ⊢ (𝐽 ∈ Top → ((int‘𝐽) Fn 𝒫 𝑋 ↔ (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)) Fn 𝒫 𝑋))
13	9, 12	mpbird 260	. . 3 ⊢ (𝐽 ∈ Top → (int‘𝐽) Fn 𝒫 𝑋)
14		elpwi 4571	. . . . 5 ⊢ (𝑠 ∈ 𝒫 𝑋 → 𝑠 ⊆ 𝑋)
15	10	ntropn 23171	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑠 ⊆ 𝑋) → ((int‘𝐽)‘𝑠) ∈ 𝐽)
16	15	ex 417	. . . . 5 ⊢ (𝐽 ∈ Top → (𝑠 ⊆ 𝑋 → ((int‘𝐽)‘𝑠) ∈ 𝐽))
17	14, 16	syl5 35	. . . 4 ⊢ (𝐽 ∈ Top → (𝑠 ∈ 𝒫 𝑋 → ((int‘𝐽)‘𝑠) ∈ 𝐽))
18	17	ralrimiv 3162	. . 3 ⊢ (𝐽 ∈ Top → ∀𝑠 ∈ 𝒫 𝑋((int‘𝐽)‘𝑠) ∈ 𝐽)
19		fnfvrnss 7114	. . 3 ⊢ (((int‘𝐽) Fn 𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋((int‘𝐽)‘𝑠) ∈ 𝐽) → ran (int‘𝐽) ⊆ 𝐽)
20	13, 18, 19	syl2anc 595	. 2 ⊢ (𝐽 ∈ Top → ran (int‘𝐽) ⊆ 𝐽)
21	2, 20	eqsstrid 3983	1 ⊢ (𝐽 ∈ Top → ran 𝐼 ⊆ 𝐽)

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  ∀wral 3085  Vcvv 3463   ∩ cin 3912   ⊆ wss 3913  𝒫 cpw 4564  ∪ cuni 4873   ↦ cmpt 5193  ran crn 5660   Fn wfn 6528  ‘cfv 6533  Topctop 23015  intcnt 23139

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-top 23016  df-ntr 23142

This theorem is referenced by:  ntrf  44734