Theorem ntrfval 12894

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 12800	. . 3 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
3		pwexg 4166	. . 3 ⊢ (𝑋 ∈ 𝐽 → 𝒫 𝑋 ∈ V)
4		mptexg 5721	. . 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 3805	. . . . . 6 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽)
7	6, 1	eqtr4di 2221	. . . . 5 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = 𝑋)
8	7	pweqd 3571	. . . 4 ⊢ (𝑗 = 𝐽 → 𝒫 ∪ 𝑗 = 𝒫 𝑋)
9		ineq1 3321	. . . . 5 ⊢ (𝑗 = 𝐽 → (𝑗 ∩ 𝒫 𝑥) = (𝐽 ∩ 𝒫 𝑥))
10	9	unieqd 3807	. . . 4 ⊢ (𝑗 = 𝐽 → ∪ (𝑗 ∩ 𝒫 𝑥) = ∪ (𝐽 ∩ 𝒫 𝑥))
11	8, 10	mpteq12dv 4071	. . 3 ⊢ (𝑗 = 𝐽 → (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ ∪ (𝑗 ∩ 𝒫 𝑥)) = (𝑥 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑥)))
12		df-ntr 12890	. . 3 ⊢ int = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ ∪ (𝑗 ∩ 𝒫 𝑥)))
13	11, 12	fvmptg 5572	. 2 ⊢ ((𝐽 ∈ Top ∧ (𝑥 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑥)) ∈ V) → (int‘𝐽) = (𝑥 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑥)))
14	5, 13	mpdan 419	1 ⊢ (𝐽 ∈ Top → (int‘𝐽) = (𝑥 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑥)))

Syntax hints:   → wi 4   = wceq 1348   ∈ wcel 2141  Vcvv 2730   ∩ cin 3120  𝒫 cpw 3566  ∪ cuni 3796   ↦ cmpt 4050  ‘cfv 5198  Topctop 12789  intcnt 12887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-top 12790  df-ntr 12890

This theorem is referenced by:  ntrval  12904