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Theorem mrerintcl 17640
Description: The relative intersection of a set of closed sets is closed. (Contributed by Stefan O'Rear, 3-Apr-2015.)
Assertion
Ref Expression
mrerintcl ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) → (𝑋 𝑆) ∈ 𝐶)

Proof of Theorem mrerintcl
StepHypRef Expression
1 rint0 4988 . . . 4 (𝑆 = ∅ → (𝑋 𝑆) = 𝑋)
21adantl 481 . . 3 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) ∧ 𝑆 = ∅) → (𝑋 𝑆) = 𝑋)
3 mre1cl 17637 . . . 4 (𝐶 ∈ (Moore‘𝑋) → 𝑋𝐶)
43ad2antrr 726 . . 3 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) ∧ 𝑆 = ∅) → 𝑋𝐶)
52, 4eqeltrd 2841 . 2 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) ∧ 𝑆 = ∅) → (𝑋 𝑆) ∈ 𝐶)
6 simp2 1138 . . . . . 6 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶𝑆 ≠ ∅) → 𝑆𝐶)
7 mresspw 17635 . . . . . . 7 (𝐶 ∈ (Moore‘𝑋) → 𝐶 ⊆ 𝒫 𝑋)
873ad2ant1 1134 . . . . . 6 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶𝑆 ≠ ∅) → 𝐶 ⊆ 𝒫 𝑋)
96, 8sstrd 3994 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶𝑆 ≠ ∅) → 𝑆 ⊆ 𝒫 𝑋)
10 simp3 1139 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶𝑆 ≠ ∅) → 𝑆 ≠ ∅)
11 rintn0 5109 . . . . 5 ((𝑆 ⊆ 𝒫 𝑋𝑆 ≠ ∅) → (𝑋 𝑆) = 𝑆)
129, 10, 11syl2anc 584 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶𝑆 ≠ ∅) → (𝑋 𝑆) = 𝑆)
13 mreintcl 17638 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶𝑆 ≠ ∅) → 𝑆𝐶)
1412, 13eqeltrd 2841 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶𝑆 ≠ ∅) → (𝑋 𝑆) ∈ 𝐶)
15143expa 1119 . 2 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) ∧ 𝑆 ≠ ∅) → (𝑋 𝑆) ∈ 𝐶)
165, 15pm2.61dane 3029 1 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) → (𝑋 𝑆) ∈ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  cin 3950  wss 3951  c0 4333  𝒫 cpw 4600   cint 4946  cfv 6561  Moorecmre 17625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-mre 17629
This theorem is referenced by:  mreacs  17701  topmtcl  36364
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