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Theorem ntrfval 21348
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.
StepHypRef Expression
1 cldval.1 . . . 4 𝑋 = 𝐽
21topopn 21230 . . 3 (𝐽 ∈ Top → 𝑋𝐽)
3 pwexg 5128 . . 3 (𝑋𝐽 → 𝒫 𝑋 ∈ V)
4 mptexg 6808 . . 3 (𝒫 𝑋 ∈ V → (𝑥 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑥)) ∈ V)
52, 3, 43syl 18 . 2 (𝐽 ∈ Top → (𝑥 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑥)) ∈ V)
6 unieq 4716 . . . . . 6 (𝑗 = 𝐽 𝑗 = 𝐽)
76, 1syl6eqr 2826 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝑋)
87pweqd 4421 . . . 4 (𝑗 = 𝐽 → 𝒫 𝑗 = 𝒫 𝑋)
9 ineq1 4062 . . . . 5 (𝑗 = 𝐽 → (𝑗 ∩ 𝒫 𝑥) = (𝐽 ∩ 𝒫 𝑥))
109unieqd 4718 . . . 4 (𝑗 = 𝐽 (𝑗 ∩ 𝒫 𝑥) = (𝐽 ∩ 𝒫 𝑥))
118, 10mpteq12dv 5008 . . 3 (𝑗 = 𝐽 → (𝑥 ∈ 𝒫 𝑗 (𝑗 ∩ 𝒫 𝑥)) = (𝑥 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑥)))
12 df-ntr 21344 . . 3 int = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 𝑗 (𝑗 ∩ 𝒫 𝑥)))
1311, 12fvmptg 6591 . 2 ((𝐽 ∈ Top ∧ (𝑥 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑥)) ∈ V) → (int‘𝐽) = (𝑥 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑥)))
145, 13mpdan 674 1 (𝐽 ∈ Top → (int‘𝐽) = (𝑥 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1507  wcel 2050  Vcvv 3409  cin 3822  𝒫 cpw 4416   cuni 4708  cmpt 5004  cfv 6185  Topctop 21217  intcnt 21341
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-rep 5045  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-top 21218  df-ntr 21344
This theorem is referenced by:  ntrval  21360  ntrrn  39864  ntrf  39865  dssmapntrcls  39870
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