Theorem ntrval 21042

Description: The interior of a subset of a topology's base set is the union of all the open sets it includes. Definition of interior of [Munkres] p. 94. (Contributed by NM, 10-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)

Hypothesis

Ref	Expression
iscld.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
ntrval	⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) = ∪ (𝐽 ∩ 𝒫 𝑆))

Proof of Theorem ntrval

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iscld.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
2	1	ntrfval 21030	. . . 4 ⊢ (𝐽 ∈ Top → (int‘𝐽) = (𝑥 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑥)))
3	2	fveq1d 6354	. . 3 ⊢ (𝐽 ∈ Top → ((int‘𝐽)‘𝑆) = ((𝑥 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑥))‘𝑆))
4	3	adantr 472	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) = ((𝑥 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑥))‘𝑆))
5	1	topopn 20913	. . . . 5 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
6		elpw2g 4976	. . . . 5 ⊢ (𝑋 ∈ 𝐽 → (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋))
7	5, 6	syl 17	. . . 4 ⊢ (𝐽 ∈ Top → (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋))
8	7	biimpar 503	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ∈ 𝒫 𝑋)
9		inex1g 4953	. . . . 5 ⊢ (𝐽 ∈ Top → (𝐽 ∩ 𝒫 𝑆) ∈ V)
10	9	adantr 472	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝐽 ∩ 𝒫 𝑆) ∈ V)
11		uniexg 7120	. . . 4 ⊢ ((𝐽 ∩ 𝒫 𝑆) ∈ V → ∪ (𝐽 ∩ 𝒫 𝑆) ∈ V)
12	10, 11	syl 17	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∪ (𝐽 ∩ 𝒫 𝑆) ∈ V)
13		pweq 4305	. . . . . 6 ⊢ (𝑥 = 𝑆 → 𝒫 𝑥 = 𝒫 𝑆)
14	13	ineq2d 3957	. . . . 5 ⊢ (𝑥 = 𝑆 → (𝐽 ∩ 𝒫 𝑥) = (𝐽 ∩ 𝒫 𝑆))
15	14	unieqd 4598	. . . 4 ⊢ (𝑥 = 𝑆 → ∪ (𝐽 ∩ 𝒫 𝑥) = ∪ (𝐽 ∩ 𝒫 𝑆))
16		eqid 2760	. . . 4 ⊢ (𝑥 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑥)) = (𝑥 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑥))
17	15, 16	fvmptg 6442	. . 3 ⊢ ((𝑆 ∈ 𝒫 𝑋 ∧ ∪ (𝐽 ∩ 𝒫 𝑆) ∈ V) → ((𝑥 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑥))‘𝑆) = ∪ (𝐽 ∩ 𝒫 𝑆))
18	8, 12, 17	syl2anc 696	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝑥 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑥))‘𝑆) = ∪ (𝐽 ∩ 𝒫 𝑆))
19	4, 18	eqtrd 2794	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) = ∪ (𝐽 ∩ 𝒫 𝑆))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1632   ∈ wcel 2139  Vcvv 3340   ∩ cin 3714   ⊆ wss 3715  𝒫 cpw 4302  ∪ cuni 4588   ↦ cmpt 4881  ‘cfv 6049  Topctop 20900  intcnt 21023

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-top 20901  df-ntr 21026

This theorem is referenced by:  ntropn  21055  clsval2  21056  ntrss2  21063  ssntr  21064  isopn3  21072  ntreq0  21083