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Theorem mrerintcl 16465
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 4651 . . . 4 (𝑆 = ∅ → (𝑋 𝑆) = 𝑋)
21adantl 467 . . 3 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) ∧ 𝑆 = ∅) → (𝑋 𝑆) = 𝑋)
3 mre1cl 16462 . . . 4 (𝐶 ∈ (Moore‘𝑋) → 𝑋𝐶)
43ad2antrr 705 . . 3 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) ∧ 𝑆 = ∅) → 𝑋𝐶)
52, 4eqeltrd 2850 . 2 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) ∧ 𝑆 = ∅) → (𝑋 𝑆) ∈ 𝐶)
6 simp2 1131 . . . . . 6 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶𝑆 ≠ ∅) → 𝑆𝐶)
7 mresspw 16460 . . . . . . 7 (𝐶 ∈ (Moore‘𝑋) → 𝐶 ⊆ 𝒫 𝑋)
873ad2ant1 1127 . . . . . 6 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶𝑆 ≠ ∅) → 𝐶 ⊆ 𝒫 𝑋)
96, 8sstrd 3762 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶𝑆 ≠ ∅) → 𝑆 ⊆ 𝒫 𝑋)
10 simp3 1132 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶𝑆 ≠ ∅) → 𝑆 ≠ ∅)
11 rintn0 4753 . . . . 5 ((𝑆 ⊆ 𝒫 𝑋𝑆 ≠ ∅) → (𝑋 𝑆) = 𝑆)
129, 10, 11syl2anc 573 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶𝑆 ≠ ∅) → (𝑋 𝑆) = 𝑆)
13 mreintcl 16463 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶𝑆 ≠ ∅) → 𝑆𝐶)
1412, 13eqeltrd 2850 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶𝑆 ≠ ∅) → (𝑋 𝑆) ∈ 𝐶)
15143expa 1111 . 2 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) ∧ 𝑆 ≠ ∅) → (𝑋 𝑆) ∈ 𝐶)
165, 15pm2.61dane 3030 1 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) → (𝑋 𝑆) ∈ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  wne 2943  cin 3722  wss 3723  c0 4063  𝒫 cpw 4297   cint 4611  cfv 6031  Moorecmre 16450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-int 4612  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-mre 16454
This theorem is referenced by:  mreacs  16526  topmtcl  32695
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