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Theorem mreriincl 16857
Description: The relative intersection of a family of closed sets is closed. (Contributed by Stefan O'Rear, 3-Apr-2015.)
Assertion
Ref Expression
mreriincl ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) → (𝑋 𝑦𝐼 𝑆) ∈ 𝐶)
Distinct variable groups:   𝑦,𝐼   𝑦,𝑋   𝑦,𝐶
Allowed substitution hint:   𝑆(𝑦)

Proof of Theorem mreriincl
StepHypRef Expression
1 riin0 4995 . . . 4 (𝐼 = ∅ → (𝑋 𝑦𝐼 𝑆) = 𝑋)
21adantl 482 . . 3 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) ∧ 𝐼 = ∅) → (𝑋 𝑦𝐼 𝑆) = 𝑋)
3 mre1cl 16853 . . . 4 (𝐶 ∈ (Moore‘𝑋) → 𝑋𝐶)
43ad2antrr 722 . . 3 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) ∧ 𝐼 = ∅) → 𝑋𝐶)
52, 4eqeltrd 2910 . 2 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) ∧ 𝐼 = ∅) → (𝑋 𝑦𝐼 𝑆) ∈ 𝐶)
6 mress 16852 . . . . . . 7 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) → 𝑆𝑋)
76ex 413 . . . . . 6 (𝐶 ∈ (Moore‘𝑋) → (𝑆𝐶𝑆𝑋))
87ralimdv 3175 . . . . 5 (𝐶 ∈ (Moore‘𝑋) → (∀𝑦𝐼 𝑆𝐶 → ∀𝑦𝐼 𝑆𝑋))
98imp 407 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) → ∀𝑦𝐼 𝑆𝑋)
10 riinn0 4996 . . . 4 ((∀𝑦𝐼 𝑆𝑋𝐼 ≠ ∅) → (𝑋 𝑦𝐼 𝑆) = 𝑦𝐼 𝑆)
119, 10sylan 580 . . 3 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) ∧ 𝐼 ≠ ∅) → (𝑋 𝑦𝐼 𝑆) = 𝑦𝐼 𝑆)
12 simpll 763 . . . 4 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) ∧ 𝐼 ≠ ∅) → 𝐶 ∈ (Moore‘𝑋))
13 simpr 485 . . . 4 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) ∧ 𝐼 ≠ ∅) → 𝐼 ≠ ∅)
14 simplr 765 . . . 4 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) ∧ 𝐼 ≠ ∅) → ∀𝑦𝐼 𝑆𝐶)
15 mreiincl 16855 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐼 ≠ ∅ ∧ ∀𝑦𝐼 𝑆𝐶) → 𝑦𝐼 𝑆𝐶)
1612, 13, 14, 15syl3anc 1363 . . 3 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) ∧ 𝐼 ≠ ∅) → 𝑦𝐼 𝑆𝐶)
1711, 16eqeltrd 2910 . 2 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) ∧ 𝐼 ≠ ∅) → (𝑋 𝑦𝐼 𝑆) ∈ 𝐶)
185, 17pm2.61dane 3101 1 ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) → (𝑋 𝑦𝐼 𝑆) ∈ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wne 3013  wral 3135  cin 3932  wss 3933  c0 4288   ciin 4911  cfv 6348  Moorecmre 16841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-int 4868  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-mre 16845
This theorem is referenced by:  acsfn1  16920  acsfn1c  16921  acsfn2  16922  acsfn1p  19507
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