Theorem ntrfval 22948

Description: The interior function on the subsets of a topology's base set. (Contributed by NM, 10-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)

Hypothesis

Ref	Expression
cldval.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
ntrfval	⊢ (𝐽 ∈ Top → (int‘𝐽) = (𝑥 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑥)))

Distinct variable groups:   𝑥,𝐽   𝑥,𝑋

Proof of Theorem ntrfval

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cldval.1	. . . 4 ⊢ 𝑋 = ∪ 𝐽
2	1	topopn 22828	. . 3 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
3		pwexg 5382	. . 3 ⊢ (𝑋 ∈ 𝐽 → 𝒫 𝑋 ∈ V)
4		mptexg 7239	. . 3 ⊢ (𝒫 𝑋 ∈ V → (𝑥 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑥)) ∈ V)
5	2, 3, 4	3syl 18	. 2 ⊢ (𝐽 ∈ Top → (𝑥 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑥)) ∈ V)
6		unieq 4923	. . . . . 6 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽)
7	6, 1	eqtr4di 2786	. . . . 5 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = 𝑋)
8	7	pweqd 4623	. . . 4 ⊢ (𝑗 = 𝐽 → 𝒫 ∪ 𝑗 = 𝒫 𝑋)
9		ineq1 4207	. . . . 5 ⊢ (𝑗 = 𝐽 → (𝑗 ∩ 𝒫 𝑥) = (𝐽 ∩ 𝒫 𝑥))
10	9	unieqd 4925	. . . 4 ⊢ (𝑗 = 𝐽 → ∪ (𝑗 ∩ 𝒫 𝑥) = ∪ (𝐽 ∩ 𝒫 𝑥))
11	8, 10	mpteq12dv 5243	. . 3 ⊢ (𝑗 = 𝐽 → (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ ∪ (𝑗 ∩ 𝒫 𝑥)) = (𝑥 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑥)))
12		df-ntr 22944	. . 3 ⊢ int = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ ∪ (𝑗 ∩ 𝒫 𝑥)))
13	11, 12	fvmptg 7008	. 2 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑥)) ∈ V) → (int‘𝐽) = (𝑥 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑥)))
14	5, 13	mpdan 685	1 ⊢ (𝐽 ∈ Top → (int‘𝐽) = (𝑥 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑥)))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  Vcvv 3473   ∩ cin 3948  𝒫 cpw 4606  ∪ cuni 4912   ↦ cmpt 5235  ‘cfv 6553  Topctop 22815  intcnt 22941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-top 22816  df-ntr 22944

This theorem is referenced by:  ntrval  22960  ntrrn  43583  ntrf  43584  dssmapntrcls  43589