Theorem ntrval 22979

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 22967	. . . 4 ⊢ (𝐽 ∈ Top → (int‘𝐽) = (𝑥 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑥)))
3	2	fveq1d 6883	. . 3 ⊢ (𝐽 ∈ Top → ((int‘𝐽)‘𝑆) = ((𝑥 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑥))‘𝑆))
4	3	adantr 480	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) = ((𝑥 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑥))‘𝑆))
5		eqid 2736	. . 3 ⊢ (𝑥 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑥)) = (𝑥 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑥))
6		pweq 4594	. . . . 5 ⊢ (𝑥 = 𝑆 → 𝒫 𝑥 = 𝒫 𝑆)
7	6	ineq2d 4200	. . . 4 ⊢ (𝑥 = 𝑆 → (𝐽 ∩ 𝒫 𝑥) = (𝐽 ∩ 𝒫 𝑆))
8	7	unieqd 4901	. . 3 ⊢ (𝑥 = 𝑆 → ∪ (𝐽 ∩ 𝒫 𝑥) = ∪ (𝐽 ∩ 𝒫 𝑆))
9	1	topopn 22849	. . . . 5 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
10		elpw2g 5308	. . . . 5 ⊢ (𝑋 ∈ 𝐽 → (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋))
11	9, 10	syl 17	. . . 4 ⊢ (𝐽 ∈ Top → (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋))
12	11	biimpar 477	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ∈ 𝒫 𝑋)
13		inex1g 5294	. . . . 5 ⊢ (𝐽 ∈ Top → (𝐽 ∩ 𝒫 𝑆) ∈ V)
14	13	adantr 480	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝐽 ∩ 𝒫 𝑆) ∈ V)
15	14	uniexd 7741	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∪ (𝐽 ∩ 𝒫 𝑆) ∈ V)
16	5, 8, 12, 15	fvmptd3 7014	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝑥 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑥))‘𝑆) = ∪ (𝐽 ∩ 𝒫 𝑆))
17	4, 16	eqtrd 2771	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) = ∪ (𝐽 ∩ 𝒫 𝑆))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3464   ∩ cin 3930   ⊆ wss 3931  𝒫 cpw 4580  ∪ cuni 4888   ↦ cmpt 5206  ‘cfv 6536  Topctop 22836  intcnt 22960

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-top 22837  df-ntr 22963

This theorem is referenced by:  ntropn  22992  clsval2  22993  ntrss2  23000  ssntr  23001  isopn3  23009  ntreq0  23020  toplatglb  48955