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Theorem mreriincl 17641
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 5082 . . . 4 (𝐼 = ∅ → (𝑋 𝑦𝐼 𝑆) = 𝑋)
21adantl 481 . . 3 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) ∧ 𝐼 = ∅) → (𝑋 𝑦𝐼 𝑆) = 𝑋)
3 mre1cl 17637 . . . 4 (𝐶 ∈ (Moore‘𝑋) → 𝑋𝐶)
43ad2antrr 726 . . 3 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) ∧ 𝐼 = ∅) → 𝑋𝐶)
52, 4eqeltrd 2841 . 2 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) ∧ 𝐼 = ∅) → (𝑋 𝑦𝐼 𝑆) ∈ 𝐶)
6 mress 17636 . . . . . . 7 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) → 𝑆𝑋)
76ex 412 . . . . . 6 (𝐶 ∈ (Moore‘𝑋) → (𝑆𝐶𝑆𝑋))
87ralimdv 3169 . . . . 5 (𝐶 ∈ (Moore‘𝑋) → (∀𝑦𝐼 𝑆𝐶 → ∀𝑦𝐼 𝑆𝑋))
98imp 406 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) → ∀𝑦𝐼 𝑆𝑋)
10 riinn0 5083 . . . 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 767 . . . 4 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) ∧ 𝐼 ≠ ∅) → 𝐶 ∈ (Moore‘𝑋))
13 simpr 484 . . . 4 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) ∧ 𝐼 ≠ ∅) → 𝐼 ≠ ∅)
14 simplr 769 . . . 4 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) ∧ 𝐼 ≠ ∅) → ∀𝑦𝐼 𝑆𝐶)
15 mreiincl 17639 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐼 ≠ ∅ ∧ ∀𝑦𝐼 𝑆𝐶) → 𝑦𝐼 𝑆𝐶)
1612, 13, 14, 15syl3anc 1373 . . 3 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) ∧ 𝐼 ≠ ∅) → 𝑦𝐼 𝑆𝐶)
1711, 16eqeltrd 2841 . 2 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) ∧ 𝐼 ≠ ∅) → (𝑋 𝑦𝐼 𝑆) ∈ 𝐶)
185, 17pm2.61dane 3029 1 ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) → (𝑋 𝑦𝐼 𝑆) ∈ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wne 2940  wral 3061  cin 3950  wss 3951  c0 4333   ciin 4992  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-iin 4994  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:  acsfn1  17704  acsfn1c  17705  acsfn2  17706  acsfn1p  20800
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