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Theorem mrerintcl 17642
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 4993 . . . 4 (𝑆 = ∅ → (𝑋 𝑆) = 𝑋)
21adantl 481 . . 3 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) ∧ 𝑆 = ∅) → (𝑋 𝑆) = 𝑋)
3 mre1cl 17639 . . . 4 (𝐶 ∈ (Moore‘𝑋) → 𝑋𝐶)
43ad2antrr 726 . . 3 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) ∧ 𝑆 = ∅) → 𝑋𝐶)
52, 4eqeltrd 2839 . 2 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) ∧ 𝑆 = ∅) → (𝑋 𝑆) ∈ 𝐶)
6 simp2 1136 . . . . . 6 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶𝑆 ≠ ∅) → 𝑆𝐶)
7 mresspw 17637 . . . . . . 7 (𝐶 ∈ (Moore‘𝑋) → 𝐶 ⊆ 𝒫 𝑋)
873ad2ant1 1132 . . . . . 6 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶𝑆 ≠ ∅) → 𝐶 ⊆ 𝒫 𝑋)
96, 8sstrd 4006 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶𝑆 ≠ ∅) → 𝑆 ⊆ 𝒫 𝑋)
10 simp3 1137 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶𝑆 ≠ ∅) → 𝑆 ≠ ∅)
11 rintn0 5114 . . . . 5 ((𝑆 ⊆ 𝒫 𝑋𝑆 ≠ ∅) → (𝑋 𝑆) = 𝑆)
129, 10, 11syl2anc 584 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶𝑆 ≠ ∅) → (𝑋 𝑆) = 𝑆)
13 mreintcl 17640 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶𝑆 ≠ ∅) → 𝑆𝐶)
1412, 13eqeltrd 2839 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶𝑆 ≠ ∅) → (𝑋 𝑆) ∈ 𝐶)
15143expa 1117 . 2 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) ∧ 𝑆 ≠ ∅) → (𝑋 𝑆) ∈ 𝐶)
165, 15pm2.61dane 3027 1 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) → (𝑋 𝑆) ∈ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938  cin 3962  wss 3963  c0 4339  𝒫 cpw 4605   cint 4951  cfv 6563  Moorecmre 17627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-mre 17631
This theorem is referenced by:  mreacs  17703  topmtcl  36346
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