Theorem ntrfval 13493

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 13399	. . 3 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
3		pwexg 4180	. . 3 ⊢ (𝑋 ∈ 𝐽 → 𝒫 𝑋 ∈ V)
4		mptexg 5741	. . 3 ⊢ (𝒫 𝑋 ∈ V → (𝑥 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑥)) ∈ V)
5	2, 3, 4	3syl 17	. 2 ⊢ (𝐽 ∈ Top → (𝑥 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑥)) ∈ V)
6		unieq 3818	. . . . . 6 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽)
7	6, 1	eqtr4di 2228	. . . . 5 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = 𝑋)
8	7	pweqd 3580	. . . 4 ⊢ (𝑗 = 𝐽 → 𝒫 ∪ 𝑗 = 𝒫 𝑋)
9		ineq1 3329	. . . . 5 ⊢ (𝑗 = 𝐽 → (𝑗 ∩ 𝒫 𝑥) = (𝐽 ∩ 𝒫 𝑥))
10	9	unieqd 3820	. . . 4 ⊢ (𝑗 = 𝐽 → ∪ (𝑗 ∩ 𝒫 𝑥) = ∪ (𝐽 ∩ 𝒫 𝑥))
11	8, 10	mpteq12dv 4085	. . 3 ⊢ (𝑗 = 𝐽 → (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ ∪ (𝑗 ∩ 𝒫 𝑥)) = (𝑥 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑥)))
12		df-ntr 13489	. . 3 ⊢ int = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ ∪ (𝑗 ∩ 𝒫 𝑥)))
13	11, 12	fvmptg 5592	. 2 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑥)) ∈ V) → (int‘𝐽) = (𝑥 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑥)))
14	5, 13	mpdan 421	1 ⊢ (𝐽 ∈ Top → (int‘𝐽) = (𝑥 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑥)))

Syntax hints:   → wi 4   = wceq 1353   ∈ wcel 2148  Vcvv 2737   ∩ cin 3128  𝒫 cpw 3575  ∪ cuni 3809   ↦ cmpt 4064  ‘cfv 5216  Topctop 13388  intcnt 13486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-pow 4174  ax-pr 4209

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-top 13389  df-ntr 13489

This theorem is referenced by:  ntrval  13503