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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 = βˆͺ 𝐽
Assertion
Ref Expression
ntrfval (𝐽 ∈ Top β†’ (intβ€˜π½) = (π‘₯ ∈ 𝒫 𝑋 ↦ βˆͺ (𝐽 ∩ 𝒫 π‘₯)))
Distinct variable groups:   π‘₯,𝐽   π‘₯,𝑋

Proof of Theorem ntrfval
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 cldval.1 . . . 4 𝑋 = βˆͺ 𝐽
21topopn 22828 . . 3 (𝐽 ∈ Top β†’ 𝑋 ∈ 𝐽)
3 pwexg 5382 . . 3 (𝑋 ∈ 𝐽 β†’ 𝒫 𝑋 ∈ V)
4 mptexg 7239 . . 3 (𝒫 𝑋 ∈ V β†’ (π‘₯ ∈ 𝒫 𝑋 ↦ βˆͺ (𝐽 ∩ 𝒫 π‘₯)) ∈ V)
52, 3, 43syl 18 . 2 (𝐽 ∈ Top β†’ (π‘₯ ∈ 𝒫 𝑋 ↦ βˆͺ (𝐽 ∩ 𝒫 π‘₯)) ∈ V)
6 unieq 4923 . . . . . 6 (𝑗 = 𝐽 β†’ βˆͺ 𝑗 = βˆͺ 𝐽)
76, 1eqtr4di 2786 . . . . 5 (𝑗 = 𝐽 β†’ βˆͺ 𝑗 = 𝑋)
87pweqd 4623 . . . 4 (𝑗 = 𝐽 β†’ 𝒫 βˆͺ 𝑗 = 𝒫 𝑋)
9 ineq1 4207 . . . . 5 (𝑗 = 𝐽 β†’ (𝑗 ∩ 𝒫 π‘₯) = (𝐽 ∩ 𝒫 π‘₯))
109unieqd 4925 . . . 4 (𝑗 = 𝐽 β†’ βˆͺ (𝑗 ∩ 𝒫 π‘₯) = βˆͺ (𝐽 ∩ 𝒫 π‘₯))
118, 10mpteq12dv 5243 . . 3 (𝑗 = 𝐽 β†’ (π‘₯ ∈ 𝒫 βˆͺ 𝑗 ↦ βˆͺ (𝑗 ∩ 𝒫 π‘₯)) = (π‘₯ ∈ 𝒫 𝑋 ↦ βˆͺ (𝐽 ∩ 𝒫 π‘₯)))
12 df-ntr 22944 . . 3 int = (𝑗 ∈ Top ↦ (π‘₯ ∈ 𝒫 βˆͺ 𝑗 ↦ βˆͺ (𝑗 ∩ 𝒫 π‘₯)))
1311, 12fvmptg 7008 . 2 ((𝐽 ∈ Top ∧ (π‘₯ ∈ 𝒫 𝑋 ↦ βˆͺ (𝐽 ∩ 𝒫 π‘₯)) ∈ V) β†’ (intβ€˜π½) = (π‘₯ ∈ 𝒫 𝑋 ↦ βˆͺ (𝐽 ∩ 𝒫 π‘₯)))
145, 13mpdan 685 1 (𝐽 ∈ Top β†’ (intβ€˜π½) = (π‘₯ ∈ 𝒫 𝑋 ↦ βˆͺ (𝐽 ∩ 𝒫 π‘₯)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  Vcvv 3473   ∩ cin 3948  π’« cpw 4606  βˆͺ cuni 4912   ↦ cmpt 5235  β€˜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
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