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Theorem mreriincl 16927
 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 4969 . . . 4 (𝐼 = ∅ → (𝑋 𝑦𝐼 𝑆) = 𝑋)
21adantl 485 . . 3 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) ∧ 𝐼 = ∅) → (𝑋 𝑦𝐼 𝑆) = 𝑋)
3 mre1cl 16923 . . . 4 (𝐶 ∈ (Moore‘𝑋) → 𝑋𝐶)
43ad2antrr 725 . . 3 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) ∧ 𝐼 = ∅) → 𝑋𝐶)
52, 4eqeltrd 2852 . 2 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) ∧ 𝐼 = ∅) → (𝑋 𝑦𝐼 𝑆) ∈ 𝐶)
6 mress 16922 . . . . . . 7 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) → 𝑆𝑋)
76ex 416 . . . . . 6 (𝐶 ∈ (Moore‘𝑋) → (𝑆𝐶𝑆𝑋))
87ralimdv 3109 . . . . 5 (𝐶 ∈ (Moore‘𝑋) → (∀𝑦𝐼 𝑆𝐶 → ∀𝑦𝐼 𝑆𝑋))
98imp 410 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) → ∀𝑦𝐼 𝑆𝑋)
10 riinn0 4970 . . . 4 ((∀𝑦𝐼 𝑆𝑋𝐼 ≠ ∅) → (𝑋 𝑦𝐼 𝑆) = 𝑦𝐼 𝑆)
119, 10sylan 583 . . 3 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) ∧ 𝐼 ≠ ∅) → (𝑋 𝑦𝐼 𝑆) = 𝑦𝐼 𝑆)
12 simpll 766 . . . 4 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) ∧ 𝐼 ≠ ∅) → 𝐶 ∈ (Moore‘𝑋))
13 simpr 488 . . . 4 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) ∧ 𝐼 ≠ ∅) → 𝐼 ≠ ∅)
14 simplr 768 . . . 4 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) ∧ 𝐼 ≠ ∅) → ∀𝑦𝐼 𝑆𝐶)
15 mreiincl 16925 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐼 ≠ ∅ ∧ ∀𝑦𝐼 𝑆𝐶) → 𝑦𝐼 𝑆𝐶)
1612, 13, 14, 15syl3anc 1368 . . 3 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) ∧ 𝐼 ≠ ∅) → 𝑦𝐼 𝑆𝐶)
1711, 16eqeltrd 2852 . 2 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) ∧ 𝐼 ≠ ∅) → (𝑋 𝑦𝐼 𝑆) ∈ 𝐶)
185, 17pm2.61dane 3038 1 ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) → (𝑋 𝑦𝐼 𝑆) ∈ 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2951  ∀wral 3070   ∩ cin 3857   ⊆ wss 3858  ∅c0 4225  ∩ ciin 4884  ‘cfv 6335  Moorecmre 16911 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-int 4839  df-iin 4886  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-iota 6294  df-fun 6337  df-fv 6343  df-mre 16915 This theorem is referenced by:  acsfn1  16990  acsfn1c  16991  acsfn2  16992  acsfn1p  19646
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