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Theorem ntrfval 12328
 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 12234 . . 3 (𝐽 ∈ Top → 𝑋𝐽)
3 pwexg 4113 . . 3 (𝑋𝐽 → 𝒫 𝑋 ∈ V)
4 mptexg 5654 . . 3 (𝒫 𝑋 ∈ V → (𝑥 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑥)) ∈ V)
52, 3, 43syl 17 . 2 (𝐽 ∈ Top → (𝑥 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑥)) ∈ V)
6 unieq 3754 . . . . . 6 (𝑗 = 𝐽 𝑗 = 𝐽)
76, 1eqtr4di 2191 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝑋)
87pweqd 3521 . . . 4 (𝑗 = 𝐽 → 𝒫 𝑗 = 𝒫 𝑋)
9 ineq1 3276 . . . . 5 (𝑗 = 𝐽 → (𝑗 ∩ 𝒫 𝑥) = (𝐽 ∩ 𝒫 𝑥))
109unieqd 3756 . . . 4 (𝑗 = 𝐽 (𝑗 ∩ 𝒫 𝑥) = (𝐽 ∩ 𝒫 𝑥))
118, 10mpteq12dv 4019 . . 3 (𝑗 = 𝐽 → (𝑥 ∈ 𝒫 𝑗 (𝑗 ∩ 𝒫 𝑥)) = (𝑥 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑥)))
12 df-ntr 12324 . . 3 int = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 𝑗 (𝑗 ∩ 𝒫 𝑥)))
1311, 12fvmptg 5506 . 2 ((𝐽 ∈ Top ∧ (𝑥 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑥)) ∈ V) → (int‘𝐽) = (𝑥 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑥)))
145, 13mpdan 418 1 (𝐽 ∈ Top → (int‘𝐽) = (𝑥 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑥)))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1332   ∈ wcel 1481  Vcvv 2690   ∩ cin 3076  𝒫 cpw 3516  ∪ cuni 3745   ↦ cmpt 3998  ‘cfv 5132  Topctop 12223  intcnt 12321 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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4052  ax-sep 4055  ax-pow 4107  ax-pr 4140 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2692  df-sbc 2915  df-csb 3009  df-un 3081  df-in 3083  df-ss 3090  df-pw 3518  df-sn 3539  df-pr 3540  df-op 3542  df-uni 3746  df-iun 3824  df-br 3939  df-opab 3999  df-mpt 4000  df-id 4224  df-xp 4554  df-rel 4555  df-cnv 4556  df-co 4557  df-dm 4558  df-rn 4559  df-res 4560  df-ima 4561  df-iota 5097  df-fun 5134  df-fn 5135  df-f 5136  df-f1 5137  df-fo 5138  df-f1o 5139  df-fv 5140  df-top 12224  df-ntr 12324 This theorem is referenced by:  ntrval  12338
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