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Theorem mrerintcl 16026
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 4446 . . . 4 (𝑆 = ∅ → (𝑋 𝑆) = 𝑋)
21adantl 480 . . 3 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) ∧ 𝑆 = ∅) → (𝑋 𝑆) = 𝑋)
3 mre1cl 16023 . . . 4 (𝐶 ∈ (Moore‘𝑋) → 𝑋𝐶)
43ad2antrr 757 . . 3 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) ∧ 𝑆 = ∅) → 𝑋𝐶)
52, 4eqeltrd 2687 . 2 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) ∧ 𝑆 = ∅) → (𝑋 𝑆) ∈ 𝐶)
6 simp2 1054 . . . . . 6 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶𝑆 ≠ ∅) → 𝑆𝐶)
7 mresspw 16021 . . . . . . 7 (𝐶 ∈ (Moore‘𝑋) → 𝐶 ⊆ 𝒫 𝑋)
873ad2ant1 1074 . . . . . 6 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶𝑆 ≠ ∅) → 𝐶 ⊆ 𝒫 𝑋)
96, 8sstrd 3577 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶𝑆 ≠ ∅) → 𝑆 ⊆ 𝒫 𝑋)
10 simp3 1055 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶𝑆 ≠ ∅) → 𝑆 ≠ ∅)
11 rintn0 4546 . . . . 5 ((𝑆 ⊆ 𝒫 𝑋𝑆 ≠ ∅) → (𝑋 𝑆) = 𝑆)
129, 10, 11syl2anc 690 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶𝑆 ≠ ∅) → (𝑋 𝑆) = 𝑆)
13 mreintcl 16024 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶𝑆 ≠ ∅) → 𝑆𝐶)
1412, 13eqeltrd 2687 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶𝑆 ≠ ∅) → (𝑋 𝑆) ∈ 𝐶)
15143expa 1256 . 2 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) ∧ 𝑆 ≠ ∅) → (𝑋 𝑆) ∈ 𝐶)
165, 15pm2.61dane 2868 1 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) → (𝑋 𝑆) ∈ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030   = wceq 1474  wcel 1976  wne 2779  cin 3538  wss 3539  c0 3873  𝒫 cpw 4107   cint 4404  cfv 5790  Moorecmre 16011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-int 4405  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-iota 5754  df-fun 5792  df-fv 5798  df-mre 16015
This theorem is referenced by:  mreacs  16088  topmtcl  31334
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