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Theorem ntrval 22953
Description: The interior of a subset of a topology's base set is the union of all the open sets it includes. Definition of interior of [Munkres] p. 94. (Contributed by NM, 10-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
Hypothesis
Ref Expression
iscld.1 𝑋 = βˆͺ 𝐽
Assertion
Ref Expression
ntrval ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π‘†) = βˆͺ (𝐽 ∩ 𝒫 𝑆))

Proof of Theorem ntrval
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 iscld.1 . . . . 5 𝑋 = βˆͺ 𝐽
21ntrfval 22941 . . . 4 (𝐽 ∈ Top β†’ (intβ€˜π½) = (π‘₯ ∈ 𝒫 𝑋 ↦ βˆͺ (𝐽 ∩ 𝒫 π‘₯)))
32fveq1d 6899 . . 3 (𝐽 ∈ Top β†’ ((intβ€˜π½)β€˜π‘†) = ((π‘₯ ∈ 𝒫 𝑋 ↦ βˆͺ (𝐽 ∩ 𝒫 π‘₯))β€˜π‘†))
43adantr 480 . 2 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π‘†) = ((π‘₯ ∈ 𝒫 𝑋 ↦ βˆͺ (𝐽 ∩ 𝒫 π‘₯))β€˜π‘†))
5 eqid 2728 . . 3 (π‘₯ ∈ 𝒫 𝑋 ↦ βˆͺ (𝐽 ∩ 𝒫 π‘₯)) = (π‘₯ ∈ 𝒫 𝑋 ↦ βˆͺ (𝐽 ∩ 𝒫 π‘₯))
6 pweq 4617 . . . . 5 (π‘₯ = 𝑆 β†’ 𝒫 π‘₯ = 𝒫 𝑆)
76ineq2d 4212 . . . 4 (π‘₯ = 𝑆 β†’ (𝐽 ∩ 𝒫 π‘₯) = (𝐽 ∩ 𝒫 𝑆))
87unieqd 4921 . . 3 (π‘₯ = 𝑆 β†’ βˆͺ (𝐽 ∩ 𝒫 π‘₯) = βˆͺ (𝐽 ∩ 𝒫 𝑆))
91topopn 22821 . . . . 5 (𝐽 ∈ Top β†’ 𝑋 ∈ 𝐽)
10 elpw2g 5346 . . . . 5 (𝑋 ∈ 𝐽 β†’ (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 βŠ† 𝑋))
119, 10syl 17 . . . 4 (𝐽 ∈ Top β†’ (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 βŠ† 𝑋))
1211biimpar 477 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ 𝑆 ∈ 𝒫 𝑋)
13 inex1g 5319 . . . . 5 (𝐽 ∈ Top β†’ (𝐽 ∩ 𝒫 𝑆) ∈ V)
1413adantr 480 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ (𝐽 ∩ 𝒫 𝑆) ∈ V)
1514uniexd 7747 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ βˆͺ (𝐽 ∩ 𝒫 𝑆) ∈ V)
165, 8, 12, 15fvmptd3 7028 . 2 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((π‘₯ ∈ 𝒫 𝑋 ↦ βˆͺ (𝐽 ∩ 𝒫 π‘₯))β€˜π‘†) = βˆͺ (𝐽 ∩ 𝒫 𝑆))
174, 16eqtrd 2768 1 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π‘†) = βˆͺ (𝐽 ∩ 𝒫 𝑆))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  Vcvv 3471   ∩ cin 3946   βŠ† wss 3947  π’« cpw 4603  βˆͺ cuni 4908   ↦ cmpt 5231  β€˜cfv 6548  Topctop 22808  intcnt 22934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-top 22809  df-ntr 22937
This theorem is referenced by:  ntropn  22966  clsval2  22967  ntrss2  22974  ssntr  22975  isopn3  22983  ntreq0  22994  toplatglb  48012
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