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Theorem iunopn 22920
Description: The indexed union of a subset of a topology is an open set. (Contributed by NM, 5-Oct-2006.)
Assertion
Ref Expression
iunopn ((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵𝐽) → 𝑥𝐴 𝐵𝐽)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐽
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem iunopn
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfiun2g 5035 . . 3 (∀𝑥𝐴 𝐵𝐽 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
21adantl 481 . 2 ((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵𝐽) → 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
3 uniiunlem 4097 . . . 4 (∀𝑥𝐴 𝐵𝐽 → (∀𝑥𝐴 𝐵𝐽 ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ⊆ 𝐽))
43ibi 267 . . 3 (∀𝑥𝐴 𝐵𝐽 → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ⊆ 𝐽)
5 uniopn 22919 . . 3 ((𝐽 ∈ Top ∧ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ⊆ 𝐽) → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ 𝐽)
64, 5sylan2 593 . 2 ((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵𝐽) → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ 𝐽)
72, 6eqeltrd 2839 1 ((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵𝐽) → 𝑥𝐴 𝐵𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  {cab 2712  wral 3059  wrex 3068  wss 3963   cuni 4912   ciun 4996  Topctop 22915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-in 3970  df-ss 3980  df-pw 4607  df-uni 4913  df-iun 4998  df-top 22916
This theorem is referenced by:  iincld  23063  tgcn  23276  kgentopon  23562  xkococnlem  23683  qtoptop2  23723  zcld  24849  metnrmlem2  24896  cnambfre  37655
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