Theorem ntrrn 44962

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 5921	. 2 ⊢ ran 𝐼 = ran (int‘𝐽)
3		vpwex 5342	. . . . . . . 8 ⊢ 𝒫 𝑠 ∈ V
4	3	inex2 5281	. . . . . . 7 ⊢ (𝐽 ∩ 𝒫 𝑠) ∈ V
5	4	uniex 7743	. . . . . 6 ⊢ ∪ (𝐽 ∩ 𝒫 𝑠) ∈ V
6	5	rgenw 3080	. . . . 5 ⊢ ∀𝑠 ∈ 𝒫 𝑋∪ (𝐽 ∩ 𝒫 𝑠) ∈ V
7		nfcv 2922	. . . . . 6 ⊢ Ⅎ𝑠𝒫 𝑋
8	7	fnmptf 6668	. . . . 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 23249	. . . . 5 ⊢ (𝐽 ∈ Top → (int‘𝐽) = (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)))
12	11	fneq1d 6625	. . . 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 4564	. . . . 5 ⊢ (𝑠 ∈ 𝒫 𝑋 → 𝑠 ⊆ 𝑋)
15	10	ntropn 23274	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑠 ⊆ 𝑋) → ((int‘𝐽)‘𝑠) ∈ 𝐽)
16	15	ex 418	. . . . 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 3153	. . 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 596	. 2 ⊢ (𝐽 ∈ Top → ran (int‘𝐽) ⊆ 𝐽)
21	2, 20	eqsstrid 3969	1 ⊢ (𝐽 ∈ Top → ran 𝐼 ⊆ 𝐽)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ∀wral 3076  Vcvv 3450   ∩ cin 3898   ⊆ wss 3899  𝒫 cpw 4557  ∪ cuni 4867   ↦ cmpt 5186  ran crn 5656   Fn wfn 6528  ‘cfv 6533  Topctop 23118  intcnt 23242

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  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 23119  df-ntr 23245

This theorem is used by:  ntrf  44963