Theorem ntrval 22923

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 22911	. . . 4 ⊢ (𝐽 ∈ Top → (int‘𝐽) = (𝑥 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑥)))
3	2	fveq1d 6860	. . 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 2729	. . 3 ⊢ (𝑥 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑥)) = (𝑥 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑥))
6		pweq 4577	. . . . 5 ⊢ (𝑥 = 𝑆 → 𝒫 𝑥 = 𝒫 𝑆)
7	6	ineq2d 4183	. . . 4 ⊢ (𝑥 = 𝑆 → (𝐽 ∩ 𝒫 𝑥) = (𝐽 ∩ 𝒫 𝑆))
8	7	unieqd 4884	. . 3 ⊢ (𝑥 = 𝑆 → ∪ (𝐽 ∩ 𝒫 𝑥) = ∪ (𝐽 ∩ 𝒫 𝑆))
9	1	topopn 22793	. . . . 5 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
10		elpw2g 5288	. . . . 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 5274	. . . . 5 ⊢ (𝐽 ∈ Top → (𝐽 ∩ 𝒫 𝑆) ∈ V)
14	13	adantr 480	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝐽 ∩ 𝒫 𝑆) ∈ V)
15	14	uniexd 7718	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∪ (𝐽 ∩ 𝒫 𝑆) ∈ V)
16	5, 8, 12, 15	fvmptd3 6991	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝑥 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑥))‘𝑆) = ∪ (𝐽 ∩ 𝒫 𝑆))
17	4, 16	eqtrd 2764	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) = ∪ (𝐽 ∩ 𝒫 𝑆))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3447   ∩ cin 3913   ⊆ wss 3914  𝒫 cpw 4563  ∪ cuni 4871   ↦ cmpt 5188  ‘cfv 6511  Topctop 22780  intcnt 22904

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 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-top 22781  df-ntr 22907

This theorem is referenced by:  ntropn  22936  clsval2  22937  ntrss2  22944  ssntr  22945  isopn3  22953  ntreq0  22964  toplatglb  48989