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Theorem iunopn 22400
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 5034 . . 3 (∀𝑥𝐴 𝐵𝐽 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
21adantl 483 . 2 ((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵𝐽) → 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
3 uniiunlem 4085 . . . 4 (∀𝑥𝐴 𝐵𝐽 → (∀𝑥𝐴 𝐵𝐽 ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ⊆ 𝐽))
43ibi 267 . . 3 (∀𝑥𝐴 𝐵𝐽 → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ⊆ 𝐽)
5 uniopn 22399 . . 3 ((𝐽 ∈ Top ∧ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ⊆ 𝐽) → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ 𝐽)
64, 5sylan2 594 . 2 ((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵𝐽) → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ 𝐽)
72, 6eqeltrd 2834 1 ((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵𝐽) → 𝑥𝐴 𝐵𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  {cab 2710  wral 3062  wrex 3071  wss 3949   cuni 4909   ciun 4998  Topctop 22395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-in 3956  df-ss 3966  df-pw 4605  df-uni 4910  df-iun 5000  df-top 22396
This theorem is referenced by:  iincld  22543  tgcn  22756  kgentopon  23042  xkococnlem  23163  qtoptop2  23203  zcld  24329  metnrmlem2  24376  cnambfre  36536
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