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Theorem iunopn 22799
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 5033 . . 3 (∀𝑥𝐴 𝐵𝐽 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
21adantl 481 . 2 ((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵𝐽) → 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
3 uniiunlem 4082 . . . 4 (∀𝑥𝐴 𝐵𝐽 → (∀𝑥𝐴 𝐵𝐽 ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ⊆ 𝐽))
43ibi 267 . . 3 (∀𝑥𝐴 𝐵𝐽 → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ⊆ 𝐽)
5 uniopn 22798 . . 3 ((𝐽 ∈ Top ∧ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ⊆ 𝐽) → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ 𝐽)
64, 5sylan2 592 . 2 ((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵𝐽) → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ 𝐽)
72, 6eqeltrd 2829 1 ((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵𝐽) → 𝑥𝐴 𝐵𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  {cab 2705  wral 3058  wrex 3067  wss 3947   cuni 4908   ciun 4996  Topctop 22794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-in 3954  df-ss 3964  df-pw 4605  df-uni 4909  df-iun 4998  df-top 22795
This theorem is referenced by:  iincld  22942  tgcn  23155  kgentopon  23441  xkococnlem  23562  qtoptop2  23602  zcld  24728  metnrmlem2  24775  cnambfre  37141
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