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Theorem ntrfval 13603
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 13511 . . 3 (𝐽 ∈ Top β†’ 𝑋 ∈ 𝐽)
3 pwexg 4181 . . 3 (𝑋 ∈ 𝐽 β†’ 𝒫 𝑋 ∈ V)
4 mptexg 5742 . . 3 (𝒫 𝑋 ∈ V β†’ (π‘₯ ∈ 𝒫 𝑋 ↦ βˆͺ (𝐽 ∩ 𝒫 π‘₯)) ∈ V)
52, 3, 43syl 17 . 2 (𝐽 ∈ Top β†’ (π‘₯ ∈ 𝒫 𝑋 ↦ βˆͺ (𝐽 ∩ 𝒫 π‘₯)) ∈ V)
6 unieq 3819 . . . . . 6 (𝑗 = 𝐽 β†’ βˆͺ 𝑗 = βˆͺ 𝐽)
76, 1eqtr4di 2228 . . . . 5 (𝑗 = 𝐽 β†’ βˆͺ 𝑗 = 𝑋)
87pweqd 3581 . . . 4 (𝑗 = 𝐽 β†’ 𝒫 βˆͺ 𝑗 = 𝒫 𝑋)
9 ineq1 3330 . . . . 5 (𝑗 = 𝐽 β†’ (𝑗 ∩ 𝒫 π‘₯) = (𝐽 ∩ 𝒫 π‘₯))
109unieqd 3821 . . . 4 (𝑗 = 𝐽 β†’ βˆͺ (𝑗 ∩ 𝒫 π‘₯) = βˆͺ (𝐽 ∩ 𝒫 π‘₯))
118, 10mpteq12dv 4086 . . 3 (𝑗 = 𝐽 β†’ (π‘₯ ∈ 𝒫 βˆͺ 𝑗 ↦ βˆͺ (𝑗 ∩ 𝒫 π‘₯)) = (π‘₯ ∈ 𝒫 𝑋 ↦ βˆͺ (𝐽 ∩ 𝒫 π‘₯)))
12 df-ntr 13599 . . 3 int = (𝑗 ∈ Top ↦ (π‘₯ ∈ 𝒫 βˆͺ 𝑗 ↦ βˆͺ (𝑗 ∩ 𝒫 π‘₯)))
1311, 12fvmptg 5593 . 2 ((𝐽 ∈ Top ∧ (π‘₯ ∈ 𝒫 𝑋 ↦ βˆͺ (𝐽 ∩ 𝒫 π‘₯)) ∈ V) β†’ (intβ€˜π½) = (π‘₯ ∈ 𝒫 𝑋 ↦ βˆͺ (𝐽 ∩ 𝒫 π‘₯)))
145, 13mpdan 421 1 (𝐽 ∈ Top β†’ (intβ€˜π½) = (π‘₯ ∈ 𝒫 𝑋 ↦ βˆͺ (𝐽 ∩ 𝒫 π‘₯)))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   = wceq 1353   ∈ wcel 2148  Vcvv 2738   ∩ cin 3129  π’« cpw 3576  βˆͺ cuni 3810   ↦ cmpt 4065  β€˜cfv 5217  Topctop 13500  intcnt 13596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-pow 4175  ax-pr 4210
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-top 13501  df-ntr 13599
This theorem is referenced by:  ntrval  13613
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