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Theorem ntrfval 23048
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 22928 . . 3 (𝐽 ∈ Top → 𝑋𝐽)
3 pwexg 5384 . . 3 (𝑋𝐽 → 𝒫 𝑋 ∈ V)
4 mptexg 7241 . . 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 4923 . . . . . 6 (𝑗 = 𝐽 𝑗 = 𝐽)
76, 1eqtr4di 2793 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝑋)
87pweqd 4622 . . . 4 (𝑗 = 𝐽 → 𝒫 𝑗 = 𝒫 𝑋)
9 ineq1 4221 . . . . 5 (𝑗 = 𝐽 → (𝑗 ∩ 𝒫 𝑥) = (𝐽 ∩ 𝒫 𝑥))
109unieqd 4925 . . . 4 (𝑗 = 𝐽 (𝑗 ∩ 𝒫 𝑥) = (𝐽 ∩ 𝒫 𝑥))
118, 10mpteq12dv 5239 . . 3 (𝑗 = 𝐽 → (𝑥 ∈ 𝒫 𝑗 (𝑗 ∩ 𝒫 𝑥)) = (𝑥 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑥)))
12 df-ntr 23044 . . 3 int = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 𝑗 (𝑗 ∩ 𝒫 𝑥)))
1311, 12fvmptg 7014 . 2 ((𝐽 ∈ Top ∧ (𝑥 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑥)) ∈ V) → (int‘𝐽) = (𝑥 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑥)))
145, 13mpdan 687 1 (𝐽 ∈ Top → (int‘𝐽) = (𝑥 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  Vcvv 3478  cin 3962  𝒫 cpw 4605   cuni 4912  cmpt 5231  cfv 6563  Topctop 22915  intcnt 23041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-top 22916  df-ntr 23044
This theorem is referenced by:  ntrval  23060  ntrrn  44112  ntrf  44113  dssmapntrcls  44118
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