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Theorem iuncld 21629
 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 4213 . . . 4 (𝑋 ∖ (𝑋 𝑥𝐴 (𝑋𝐵))) = (𝑋 𝑥𝐴 (𝑋𝐵))
2 iundif2 4969 . . . 4 𝑥𝐴 (𝑋 ∖ (𝑋𝐵)) = (𝑋 𝑥𝐴 (𝑋𝐵))
31, 2eqtr4i 2847 . . 3 (𝑋 ∖ (𝑋 𝑥𝐴 (𝑋𝐵))) = 𝑥𝐴 (𝑋 ∖ (𝑋𝐵))
4 clscld.1 . . . . . . . 8 𝑋 = 𝐽
54cldss 21613 . . . . . . 7 (𝐵 ∈ (Clsd‘𝐽) → 𝐵𝑋)
6 dfss4 4210 . . . . . . 7 (𝐵𝑋 ↔ (𝑋 ∖ (𝑋𝐵)) = 𝐵)
75, 6sylib 221 . . . . . 6 (𝐵 ∈ (Clsd‘𝐽) → (𝑋 ∖ (𝑋𝐵)) = 𝐵)
87ralimi 3148 . . . . 5 (∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽) → ∀𝑥𝐴 (𝑋 ∖ (𝑋𝐵)) = 𝐵)
983ad2ant3 1132 . . . 4 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∀𝑥𝐴 (𝑋 ∖ (𝑋𝐵)) = 𝐵)
10 iuneq2 4911 . . . 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 2868 . 2 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → (𝑋 ∖ (𝑋 𝑥𝐴 (𝑋𝐵))) = 𝑥𝐴 𝐵)
13 simp1 1133 . . 3 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝐽 ∈ Top)
144cldopn 21615 . . . . 5 (𝐵 ∈ (Clsd‘𝐽) → (𝑋𝐵) ∈ 𝐽)
1514ralimi 3148 . . . 4 (∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽) → ∀𝑥𝐴 (𝑋𝐵) ∈ 𝐽)
164riinopn 21492 . . . 4 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 (𝑋𝐵) ∈ 𝐽) → (𝑋 𝑥𝐴 (𝑋𝐵)) ∈ 𝐽)
1715, 16syl3an3 1162 . . 3 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → (𝑋 𝑥𝐴 (𝑋𝐵)) ∈ 𝐽)
184opncld 21617 . . 3 ((𝐽 ∈ Top ∧ (𝑋 𝑥𝐴 (𝑋𝐵)) ∈ 𝐽) → (𝑋 ∖ (𝑋 𝑥𝐴 (𝑋𝐵))) ∈ (Clsd‘𝐽))
1913, 17, 18syl2anc 587 . 2 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → (𝑋 ∖ (𝑋 𝑥𝐴 (𝑋𝐵))) ∈ (Clsd‘𝐽))
2012, 19eqeltrrd 2913 1 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝑥𝐴 𝐵 ∈ (Clsd‘𝐽))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115  ∀wral 3126   ∖ cdif 3907   ∩ cin 3909   ⊆ wss 3910  ∪ cuni 4811  ∪ ciun 4892  ∩ ciin 4893  ‘cfv 6328  Fincfn 8484  Topctop 21477  Clsdccld 21600 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-int 4850  df-iun 4894  df-iin 4895  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-om 7556  df-1st 7664  df-2nd 7665  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-1o 8077  df-oadd 8081  df-er 8264  df-en 8485  df-dom 8486  df-fin 8488  df-top 21478  df-cld 21603 This theorem is referenced by:  unicld  21630  t1ficld  21911  mblfinlem1  34980  mblfinlem2  34981
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