Theorem ntrrn 44126

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 5952	. 2 ⊢ ran 𝐼 = ran (int‘𝐽)
3		vpwex 5384	. . . . . . . 8 ⊢ 𝒫 𝑠 ∈ V
4	3	inex2 5325	. . . . . . 7 ⊢ (𝐽 ∩ 𝒫 𝑠) ∈ V
5	4	uniex 7764	. . . . . 6 ⊢ ∪ (𝐽 ∩ 𝒫 𝑠) ∈ V
6	5	rgenw 3064	. . . . 5 ⊢ ∀𝑠 ∈ 𝒫 𝑋∪ (𝐽 ∩ 𝒫 𝑠) ∈ V
7		nfcv 2904	. . . . . 6 ⊢ Ⅎ𝑠𝒫 𝑋
8	7	fnmptf 6709	. . . . 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 23054	. . . . 5 ⊢ (𝐽 ∈ Top → (int‘𝐽) = (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)))
12	11	fneq1d 6666	. . . 4 ⊢ (𝐽 ∈ Top → ((int‘𝐽) Fn 𝒫 𝑋 ↔ (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)) Fn 𝒫 𝑋))
13	9, 12	mpbird 257	. . 3 ⊢ (𝐽 ∈ Top → (int‘𝐽) Fn 𝒫 𝑋)
14		elpwi 4613	. . . . 5 ⊢ (𝑠 ∈ 𝒫 𝑋 → 𝑠 ⊆ 𝑋)
15	10	ntropn 23079	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑠 ⊆ 𝑋) → ((int‘𝐽)‘𝑠) ∈ 𝐽)
16	15	ex 412	. . . . 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 3144	. . 3 ⊢ (𝐽 ∈ Top → ∀𝑠 ∈ 𝒫 𝑋((int‘𝐽)‘𝑠) ∈ 𝐽)
19		fnfvrnss 7145	. . 3 ⊢ (((int‘𝐽) Fn 𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋((int‘𝐽)‘𝑠) ∈ 𝐽) → ran (int‘𝐽) ⊆ 𝐽)
20	13, 18, 19	syl2anc 584	. 2 ⊢ (𝐽 ∈ Top → ran (int‘𝐽) ⊆ 𝐽)
21	2, 20	eqsstrid 4045	1 ⊢ (𝐽 ∈ Top → ran 𝐼 ⊆ 𝐽)

Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2107  ∀wral 3060  Vcvv 3479   ∩ cin 3963   ⊆ wss 3964  𝒫 cpw 4606  ∪ cuni 4913   ↦ cmpt 5232  ran crn 5691   Fn wfn 6561  ‘cfv 6566  Topctop 22921  intcnt 23047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5286  ax-sep 5303  ax-nul 5313  ax-pow 5372  ax-pr 5439  ax-un 7758

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1541  df-fal 1551  df-ex 1778  df-nf 1782  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3435  df-v 3481  df-sbc 3793  df-csb 3910  df-dif 3967  df-un 3969  df-in 3971  df-ss 3981  df-nul 4341  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4914  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5584  df-xp 5696  df-rel 5697  df-cnv 5698  df-co 5699  df-dm 5700  df-rn 5701  df-res 5702  df-ima 5703  df-iota 6519  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-top 22922  df-ntr 23050

This theorem is referenced by:  ntrf  44127