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Theorem ntrval 22403
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 22391 . . . 4 (𝐽 ∈ Top β†’ (intβ€˜π½) = (π‘₯ ∈ 𝒫 𝑋 ↦ βˆͺ (𝐽 ∩ 𝒫 π‘₯)))
32fveq1d 6845 . . 3 (𝐽 ∈ Top β†’ ((intβ€˜π½)β€˜π‘†) = ((π‘₯ ∈ 𝒫 𝑋 ↦ βˆͺ (𝐽 ∩ 𝒫 π‘₯))β€˜π‘†))
43adantr 482 . 2 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π‘†) = ((π‘₯ ∈ 𝒫 𝑋 ↦ βˆͺ (𝐽 ∩ 𝒫 π‘₯))β€˜π‘†))
5 eqid 2733 . . 3 (π‘₯ ∈ 𝒫 𝑋 ↦ βˆͺ (𝐽 ∩ 𝒫 π‘₯)) = (π‘₯ ∈ 𝒫 𝑋 ↦ βˆͺ (𝐽 ∩ 𝒫 π‘₯))
6 pweq 4575 . . . . 5 (π‘₯ = 𝑆 β†’ 𝒫 π‘₯ = 𝒫 𝑆)
76ineq2d 4173 . . . 4 (π‘₯ = 𝑆 β†’ (𝐽 ∩ 𝒫 π‘₯) = (𝐽 ∩ 𝒫 𝑆))
87unieqd 4880 . . 3 (π‘₯ = 𝑆 β†’ βˆͺ (𝐽 ∩ 𝒫 π‘₯) = βˆͺ (𝐽 ∩ 𝒫 𝑆))
91topopn 22271 . . . . 5 (𝐽 ∈ Top β†’ 𝑋 ∈ 𝐽)
10 elpw2g 5302 . . . . 5 (𝑋 ∈ 𝐽 β†’ (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 βŠ† 𝑋))
119, 10syl 17 . . . 4 (𝐽 ∈ Top β†’ (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 βŠ† 𝑋))
1211biimpar 479 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ 𝑆 ∈ 𝒫 𝑋)
13 inex1g 5277 . . . . 5 (𝐽 ∈ Top β†’ (𝐽 ∩ 𝒫 𝑆) ∈ V)
1413adantr 482 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ (𝐽 ∩ 𝒫 𝑆) ∈ V)
1514uniexd 7680 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ βˆͺ (𝐽 ∩ 𝒫 𝑆) ∈ V)
165, 8, 12, 15fvmptd3 6972 . 2 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((π‘₯ ∈ 𝒫 𝑋 ↦ βˆͺ (𝐽 ∩ 𝒫 π‘₯))β€˜π‘†) = βˆͺ (𝐽 ∩ 𝒫 𝑆))
174, 16eqtrd 2773 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 397   = wceq 1542   ∈ wcel 2107  Vcvv 3444   ∩ cin 3910   βŠ† wss 3911  π’« cpw 4561  βˆͺ cuni 4866   ↦ cmpt 5189  β€˜cfv 6497  Topctop 22258  intcnt 22384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-top 22259  df-ntr 22387
This theorem is referenced by:  ntropn  22416  clsval2  22417  ntrss2  22424  ssntr  22425  isopn3  22433  ntreq0  22444  toplatglb  47112
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