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

Theorem riincld 21069
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 4728 . . . 4 (𝐴 = ∅ → (𝑋 𝑥𝐴 𝐵) = 𝑋)
21adantl 467 . . 3 (((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝐴 = ∅) → (𝑋 𝑥𝐴 𝐵) = 𝑋)
3 clscld.1 . . . . 5 𝑋 = 𝐽
43topcld 21060 . . . 4 (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
54ad2antrr 705 . . 3 (((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝐴 = ∅) → 𝑋 ∈ (Clsd‘𝐽))
62, 5eqeltrd 2850 . 2 (((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝐴 = ∅) → (𝑋 𝑥𝐴 𝐵) ∈ (Clsd‘𝐽))
74ad2antrr 705 . . 3 (((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝐴 ≠ ∅) → 𝑋 ∈ (Clsd‘𝐽))
8 simpr 471 . . . 4 (((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝐴 ≠ ∅) → 𝐴 ≠ ∅)
9 simplr 752 . . . 4 (((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝐴 ≠ ∅) → ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽))
10 iincld 21064 . . . 4 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝑥𝐴 𝐵 ∈ (Clsd‘𝐽))
118, 9, 10syl2anc 573 . . 3 (((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝐴 ≠ ∅) → 𝑥𝐴 𝐵 ∈ (Clsd‘𝐽))
12 incld 21068 . . 3 ((𝑋 ∈ (Clsd‘𝐽) ∧ 𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → (𝑋 𝑥𝐴 𝐵) ∈ (Clsd‘𝐽))
137, 11, 12syl2anc 573 . 2 (((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝐴 ≠ ∅) → (𝑋 𝑥𝐴 𝐵) ∈ (Clsd‘𝐽))
146, 13pm2.61dane 3030 1 ((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → (𝑋 𝑥𝐴 𝐵) ∈ (Clsd‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wne 2943  wral 3061  cin 3722  c0 4063   cuni 4574   ciin 4655  cfv 6031  Topctop 20918  Clsdccld 21041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-iin 4657  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fn 6034  df-fv 6039  df-top 20919  df-cld 21044
This theorem is referenced by:  ptcld  21637  csscld  23267
  Copyright terms: Public domain W3C validator