Theorem ntrrn 39978

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 5696	. 2 ⊢ ran 𝐼 = ran (int‘𝐽)
3		vpwex 5176	. . . . . . . 8 ⊢ 𝒫 𝑠 ∈ V
4	3	inex2 5120	. . . . . . 7 ⊢ (𝐽 ∩ 𝒫 𝑠) ∈ V
5	4	uniex 7330	. . . . . 6 ⊢ ∪ (𝐽 ∩ 𝒫 𝑠) ∈ V
6	5	rgenw 3119	. . . . 5 ⊢ ∀𝑠 ∈ 𝒫 𝑋∪ (𝐽 ∩ 𝒫 𝑠) ∈ V
7		nfcv 2951	. . . . . 6 ⊢ Ⅎ𝑠𝒫 𝑋
8	7	fnmptf 6359	. . . . 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 21320	. . . . 5 ⊢ (𝐽 ∈ Top → (int‘𝐽) = (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)))
12	11	fneq1d 6323	. . . 4 ⊢ (𝐽 ∈ Top → ((int‘𝐽) Fn 𝒫 𝑋 ↔ (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)) Fn 𝒫 𝑋))
13	9, 12	mpbird 258	. . 3 ⊢ (𝐽 ∈ Top → (int‘𝐽) Fn 𝒫 𝑋)
14		elpwi 4469	. . . . 5 ⊢ (𝑠 ∈ 𝒫 𝑋 → 𝑠 ⊆ 𝑋)
15	10	ntropn 21345	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑠 ⊆ 𝑋) → ((int‘𝐽)‘𝑠) ∈ 𝐽)
16	15	ex 413	. . . . 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 3150	. . 3 ⊢ (𝐽 ∈ Top → ∀𝑠 ∈ 𝒫 𝑋((int‘𝐽)‘𝑠) ∈ 𝐽)
19		fnfvrnss 6754	. . 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 3942	1 ⊢ (𝐽 ∈ Top → ran 𝐼 ⊆ 𝐽)

Syntax hints:   → wi 4   = wceq 1525   ∈ wcel 2083  ∀wral 3107  Vcvv 3440   ∩ cin 3864   ⊆ wss 3865  𝒫 cpw 4459  ∪ cuni 4751   ↦ cmpt 5047  ran crn 5451   Fn wfn 6227  ‘cfv 6232  Topctop 21189  intcnt 21313

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-top 21190  df-ntr 21316

This theorem is referenced by:  ntrf  39979