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Theorem mreriincl 17432
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 5040 . . . 4 (𝐼 = ∅ → (𝑋 𝑦𝐼 𝑆) = 𝑋)
21adantl 482 . . 3 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) ∧ 𝐼 = ∅) → (𝑋 𝑦𝐼 𝑆) = 𝑋)
3 mre1cl 17428 . . . 4 (𝐶 ∈ (Moore‘𝑋) → 𝑋𝐶)
43ad2antrr 724 . . 3 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) ∧ 𝐼 = ∅) → 𝑋𝐶)
52, 4eqeltrd 2838 . 2 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) ∧ 𝐼 = ∅) → (𝑋 𝑦𝐼 𝑆) ∈ 𝐶)
6 mress 17427 . . . . . . 7 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) → 𝑆𝑋)
76ex 413 . . . . . 6 (𝐶 ∈ (Moore‘𝑋) → (𝑆𝐶𝑆𝑋))
87ralimdv 3164 . . . . 5 (𝐶 ∈ (Moore‘𝑋) → (∀𝑦𝐼 𝑆𝐶 → ∀𝑦𝐼 𝑆𝑋))
98imp 407 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) → ∀𝑦𝐼 𝑆𝑋)
10 riinn0 5041 . . . 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 765 . . . 4 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) ∧ 𝐼 ≠ ∅) → 𝐶 ∈ (Moore‘𝑋))
13 simpr 485 . . . 4 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) ∧ 𝐼 ≠ ∅) → 𝐼 ≠ ∅)
14 simplr 767 . . . 4 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) ∧ 𝐼 ≠ ∅) → ∀𝑦𝐼 𝑆𝐶)
15 mreiincl 17430 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐼 ≠ ∅ ∧ ∀𝑦𝐼 𝑆𝐶) → 𝑦𝐼 𝑆𝐶)
1612, 13, 14, 15syl3anc 1371 . . 3 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) ∧ 𝐼 ≠ ∅) → 𝑦𝐼 𝑆𝐶)
1711, 16eqeltrd 2838 . 2 (((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) ∧ 𝐼 ≠ ∅) → (𝑋 𝑦𝐼 𝑆) ∈ 𝐶)
185, 17pm2.61dane 3030 1 ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) → (𝑋 𝑦𝐼 𝑆) ∈ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wne 2941  wral 3062  cin 3907  wss 3908  c0 4280   ciin 4953  cfv 6493  Moorecmre 17416
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-int 4906  df-iin 4955  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6445  df-fun 6495  df-fv 6501  df-mre 17420
This theorem is referenced by:  acsfn1  17495  acsfn1c  17496  acsfn2  17497  acsfn1p  20213
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