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

Theorem iuncld 21071
Description: A finite indexed union of closed sets is closed. (Contributed by Mario Carneiro, 19-Sep-2015.)
Hypothesis
Ref Expression
clscld.1 𝑋 = 𝐽
Assertion
Ref Expression
iuncld ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝑥𝐴 𝐵 ∈ (Clsd‘𝐽))
Distinct variable groups:   𝑥,𝐽   𝑥,𝑋   𝑥,𝐴
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem iuncld
StepHypRef Expression
1 difin 4004 . . . 4 (𝑋 ∖ (𝑋 𝑥𝐴 (𝑋𝐵))) = (𝑋 𝑥𝐴 (𝑋𝐵))
2 iundif2 4739 . . . 4 𝑥𝐴 (𝑋 ∖ (𝑋𝐵)) = (𝑋 𝑥𝐴 (𝑋𝐵))
31, 2eqtr4i 2785 . . 3 (𝑋 ∖ (𝑋 𝑥𝐴 (𝑋𝐵))) = 𝑥𝐴 (𝑋 ∖ (𝑋𝐵))
4 clscld.1 . . . . . . . 8 𝑋 = 𝐽
54cldss 21055 . . . . . . 7 (𝐵 ∈ (Clsd‘𝐽) → 𝐵𝑋)
6 dfss4 4001 . . . . . . 7 (𝐵𝑋 ↔ (𝑋 ∖ (𝑋𝐵)) = 𝐵)
75, 6sylib 208 . . . . . 6 (𝐵 ∈ (Clsd‘𝐽) → (𝑋 ∖ (𝑋𝐵)) = 𝐵)
87ralimi 3090 . . . . 5 (∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽) → ∀𝑥𝐴 (𝑋 ∖ (𝑋𝐵)) = 𝐵)
983ad2ant3 1130 . . . 4 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∀𝑥𝐴 (𝑋 ∖ (𝑋𝐵)) = 𝐵)
10 iuneq2 4689 . . . 4 (∀𝑥𝐴 (𝑋 ∖ (𝑋𝐵)) = 𝐵 𝑥𝐴 (𝑋 ∖ (𝑋𝐵)) = 𝑥𝐴 𝐵)
119, 10syl 17 . . 3 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝑥𝐴 (𝑋 ∖ (𝑋𝐵)) = 𝑥𝐴 𝐵)
123, 11syl5eq 2806 . 2 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → (𝑋 ∖ (𝑋 𝑥𝐴 (𝑋𝐵))) = 𝑥𝐴 𝐵)
13 simp1 1131 . . 3 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝐽 ∈ Top)
144cldopn 21057 . . . . 5 (𝐵 ∈ (Clsd‘𝐽) → (𝑋𝐵) ∈ 𝐽)
1514ralimi 3090 . . . 4 (∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽) → ∀𝑥𝐴 (𝑋𝐵) ∈ 𝐽)
164riinopn 20935 . . . 4 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝑋𝐵) ∈ 𝐽) → (𝑋 𝑥𝐴 (𝑋𝐵)) ∈ 𝐽)
1715, 16syl3an3 1170 . . 3 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → (𝑋 𝑥𝐴 (𝑋𝐵)) ∈ 𝐽)
184opncld 21059 . . 3 ((𝐽 ∈ Top ∧ (𝑋 𝑥𝐴 (𝑋𝐵)) ∈ 𝐽) → (𝑋 ∖ (𝑋 𝑥𝐴 (𝑋𝐵))) ∈ (Clsd‘𝐽))
1913, 17, 18syl2anc 696 . 2 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → (𝑋 ∖ (𝑋 𝑥𝐴 (𝑋𝐵))) ∈ (Clsd‘𝐽))
2012, 19eqeltrrd 2840 1 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝑥𝐴 𝐵 ∈ (Clsd‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1072   = wceq 1632  wcel 2139  wral 3050  cdif 3712  cin 3714  wss 3715   cuni 4588   ciun 4672   ciin 4673  cfv 6049  Fincfn 8123  Topctop 20920  Clsdccld 21042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-oadd 7734  df-er 7913  df-en 8124  df-dom 8125  df-fin 8127  df-top 20921  df-cld 21045
This theorem is referenced by:  unicld  21072  t1ficld  21353  mblfinlem1  33777  mblfinlem2  33778
  Copyright terms: Public domain W3C validator