Theorem ntrrn 41409

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 5806	. 2 ⊢ ran 𝐼 = ran (int‘𝐽)
3		vpwex 5270	. . . . . . . 8 ⊢ 𝒫 𝑠 ∈ V
4	3	inex2 5211	. . . . . . 7 ⊢ (𝐽 ∩ 𝒫 𝑠) ∈ V
5	4	uniex 7529	. . . . . 6 ⊢ ∪ (𝐽 ∩ 𝒫 𝑠) ∈ V
6	5	rgenw 3073	. . . . 5 ⊢ ∀𝑠 ∈ 𝒫 𝑋∪ (𝐽 ∩ 𝒫 𝑠) ∈ V
7		nfcv 2904	. . . . . 6 ⊢ Ⅎ𝑠𝒫 𝑋
8	7	fnmptf 6514	. . . . 5 ⊢ (∀𝑠 ∈ 𝒫 𝑋∪ (𝐽 ∩ 𝒫 𝑠) ∈ V → (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)) Fn 𝒫 𝑋)
9	6, 8	mp1i 13	. . . 4 ⊢ (𝐽 ∈ Top → (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)) Fn 𝒫 𝑋)
10		ntrrn.x	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
11	10	ntrfval 21921	. . . . 5 ⊢ (𝐽 ∈ Top → (int‘𝐽) = (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)))
12	11	fneq1d 6472	. . . 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 4522	. . . . 5 ⊢ (𝑠 ∈ 𝒫 𝑋 → 𝑠 ⊆ 𝑋)
15	10	ntropn 21946	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑠 ⊆ 𝑋) → ((int‘𝐽)‘𝑠) ∈ 𝐽)
16	15	ex 416	. . . . 5 ⊢ (𝐽 ∈ Top → (𝑠 ⊆ 𝑋 → ((int‘𝐽)‘𝑠) ∈ 𝐽))
17	14, 16	syl5 34	. . . 4 ⊢ (𝐽 ∈ Top → (𝑠 ∈ 𝒫 𝑋 → ((int‘𝐽)‘𝑠) ∈ 𝐽))
18	17	ralrimiv 3104	. . 3 ⊢ (𝐽 ∈ Top → ∀𝑠 ∈ 𝒫 𝑋((int‘𝐽)‘𝑠) ∈ 𝐽)
19		fnfvrnss 6937	. . 3 ⊢ (((int‘𝐽) Fn 𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋((int‘𝐽)‘𝑠) ∈ 𝐽) → ran (int‘𝐽) ⊆ 𝐽)
20	13, 18, 19	syl2anc 587	. 2 ⊢ (𝐽 ∈ Top → ran (int‘𝐽) ⊆ 𝐽)
21	2, 20	eqsstrid 3949	1 ⊢ (𝐽 ∈ Top → ran 𝐼 ⊆ 𝐽)

Syntax hints:   → wi 4   = wceq 1543   ∈ wcel 2110  ∀wral 3061  Vcvv 3408   ∩ cin 3865   ⊆ wss 3866  𝒫 cpw 4513  ∪ cuni 4819   ↦ cmpt 5135  ran crn 5552   Fn wfn 6375  ‘cfv 6380  Topctop 21790  intcnt 21914

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-top 21791  df-ntr 21917

This theorem is referenced by:  ntrf  41410