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Theorem iunopn 21110
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 4785 . . 3 (∀𝑥𝐴 𝐵𝐽 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
21adantl 475 . 2 ((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵𝐽) → 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
3 uniiunlem 3913 . . . 4 (∀𝑥𝐴 𝐵𝐽 → (∀𝑥𝐴 𝐵𝐽 ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ⊆ 𝐽))
43ibi 259 . . 3 (∀𝑥𝐴 𝐵𝐽 → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ⊆ 𝐽)
5 uniopn 21109 . . 3 ((𝐽 ∈ Top ∧ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ⊆ 𝐽) → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ 𝐽)
64, 5sylan2 586 . 2 ((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵𝐽) → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ 𝐽)
72, 6eqeltrd 2859 1 ((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵𝐽) → 𝑥𝐴 𝐵𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1601  wcel 2107  {cab 2763  wral 3090  wrex 3091  wss 3792   cuni 4671   ciun 4753  Topctop 21105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-v 3400  df-in 3799  df-ss 3806  df-pw 4381  df-uni 4672  df-iun 4755  df-top 21106
This theorem is referenced by:  iincld  21251  tgcn  21464  kgentopon  21750  xkococnlem  21871  qtoptop2  21911  zcld  23024  metnrmlem2  23071  cnambfre  34083
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