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Theorem mreriincl 16081
 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 4530 . . . 4 (𝐼 = ∅ → (𝑋 𝑦𝐼 𝑆) = 𝑋)
21adantl 481 . . 3 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) ∧ 𝐼 = ∅) → (𝑋 𝑦𝐼 𝑆) = 𝑋)
3 mre1cl 16077 . . . 4 (𝐶 ∈ (Moore‘𝑋) → 𝑋𝐶)
43ad2antrr 758 . . 3 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) ∧ 𝐼 = ∅) → 𝑋𝐶)
52, 4eqeltrd 2688 . 2 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) ∧ 𝐼 = ∅) → (𝑋 𝑦𝐼 𝑆) ∈ 𝐶)
6 mress 16076 . . . . . . 7 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) → 𝑆𝑋)
76ex 449 . . . . . 6 (𝐶 ∈ (Moore‘𝑋) → (𝑆𝐶𝑆𝑋))
87ralimdv 2946 . . . . 5 (𝐶 ∈ (Moore‘𝑋) → (∀𝑦𝐼 𝑆𝐶 → ∀𝑦𝐼 𝑆𝑋))
98imp 444 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) → ∀𝑦𝐼 𝑆𝑋)
10 riinn0 4531 . . . 4 ((∀𝑦𝐼 𝑆𝑋𝐼 ≠ ∅) → (𝑋 𝑦𝐼 𝑆) = 𝑦𝐼 𝑆)
119, 10sylan 487 . . 3 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) ∧ 𝐼 ≠ ∅) → (𝑋 𝑦𝐼 𝑆) = 𝑦𝐼 𝑆)
12 simpll 786 . . . 4 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) ∧ 𝐼 ≠ ∅) → 𝐶 ∈ (Moore‘𝑋))
13 simpr 476 . . . 4 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) ∧ 𝐼 ≠ ∅) → 𝐼 ≠ ∅)
14 simplr 788 . . . 4 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) ∧ 𝐼 ≠ ∅) → ∀𝑦𝐼 𝑆𝐶)
15 mreiincl 16079 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐼 ≠ ∅ ∧ ∀𝑦𝐼 𝑆𝐶) → 𝑦𝐼 𝑆𝐶)
1612, 13, 14, 15syl3anc 1318 . . 3 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) ∧ 𝐼 ≠ ∅) → 𝑦𝐼 𝑆𝐶)
1711, 16eqeltrd 2688 . 2 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) ∧ 𝐼 ≠ ∅) → (𝑋 𝑦𝐼 𝑆) ∈ 𝐶)
185, 17pm2.61dane 2869 1 ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) → (𝑋 𝑦𝐼 𝑆) ∈ 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  ∩ ciin 4456  ‘cfv 5804  Moorecmre 16065 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-mre 16069 This theorem is referenced by:  acsfn1  16145  acsfn1c  16146  acsfn2  16147  acsfn1p  36788
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