Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  riincld Structured version   Visualization version   GIF version

Theorem riincld 20842
 Description: An indexed relative intersection of closed sets is closed. (Contributed by Stefan O'Rear, 22-Feb-2015.)
Hypothesis
Ref Expression
clscld.1 𝑋 = 𝐽
Assertion
Ref Expression
riincld ((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → (𝑋 𝑥𝐴 𝐵) ∈ (Clsd‘𝐽))
Distinct variable groups:   𝑥,𝐽   𝑥,𝑋   𝑥,𝐴
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem riincld
StepHypRef Expression
1 riin0 4592 . . . 4 (𝐴 = ∅ → (𝑋 𝑥𝐴 𝐵) = 𝑋)
21adantl 482 . . 3 (((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝐴 = ∅) → (𝑋 𝑥𝐴 𝐵) = 𝑋)
3 clscld.1 . . . . 5 𝑋 = 𝐽
43topcld 20833 . . . 4 (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
54ad2antrr 762 . . 3 (((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝐴 = ∅) → 𝑋 ∈ (Clsd‘𝐽))
62, 5eqeltrd 2700 . 2 (((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝐴 = ∅) → (𝑋 𝑥𝐴 𝐵) ∈ (Clsd‘𝐽))
74ad2antrr 762 . . 3 (((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝐴 ≠ ∅) → 𝑋 ∈ (Clsd‘𝐽))
8 simpr 477 . . . 4 (((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝐴 ≠ ∅) → 𝐴 ≠ ∅)
9 simplr 792 . . . 4 (((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝐴 ≠ ∅) → ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽))
10 iincld 20837 . . . 4 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝑥𝐴 𝐵 ∈ (Clsd‘𝐽))
118, 9, 10syl2anc 693 . . 3 (((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝐴 ≠ ∅) → 𝑥𝐴 𝐵 ∈ (Clsd‘𝐽))
12 incld 20841 . . 3 ((𝑋 ∈ (Clsd‘𝐽) ∧ 𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → (𝑋 𝑥𝐴 𝐵) ∈ (Clsd‘𝐽))
137, 11, 12syl2anc 693 . 2 (((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝐴 ≠ ∅) → (𝑋 𝑥𝐴 𝐵) ∈ (Clsd‘𝐽))
146, 13pm2.61dane 2880 1 ((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → (𝑋 𝑥𝐴 𝐵) ∈ (Clsd‘𝐽))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1482   ∈ wcel 1989   ≠ wne 2793  ∀wral 2911   ∩ cin 3571  ∅c0 3913  ∪ cuni 4434  ∩ ciin 4519  ‘cfv 5886  Topctop 20692  Clsdccld 20814 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-int 4474  df-iun 4520  df-iin 4521  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-iota 5849  df-fun 5888  df-fn 5889  df-fv 5894  df-top 20693  df-cld 20817 This theorem is referenced by:  ptcld  21410  csscld  23042
 Copyright terms: Public domain W3C validator