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Theorem uniopn 22819
Description: The union of a subset of a topology (that is, the union of any family of open sets of a topology) is an open set. (Contributed by Stefan Allan, 27-Feb-2006.)
Assertion
Ref Expression
uniopn ((𝐽 ∈ Top ∧ 𝐴𝐽) → 𝐴𝐽)

Proof of Theorem uniopn
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 istopg 22817 . . . . 5 (𝐽 ∈ Top → (𝐽 ∈ Top ↔ (∀𝑥(𝑥𝐽 𝑥𝐽) ∧ ∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽)))
21ibi 266 . . . 4 (𝐽 ∈ Top → (∀𝑥(𝑥𝐽 𝑥𝐽) ∧ ∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽))
32simpld 493 . . 3 (𝐽 ∈ Top → ∀𝑥(𝑥𝐽 𝑥𝐽))
4 elpw2g 5350 . . . . . . . 8 (𝐽 ∈ Top → (𝐴 ∈ 𝒫 𝐽𝐴𝐽))
54biimpar 476 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐴𝐽) → 𝐴 ∈ 𝒫 𝐽)
6 sseq1 4007 . . . . . . . . 9 (𝑥 = 𝐴 → (𝑥𝐽𝐴𝐽))
7 unieq 4923 . . . . . . . . . 10 (𝑥 = 𝐴 𝑥 = 𝐴)
87eleq1d 2814 . . . . . . . . 9 (𝑥 = 𝐴 → ( 𝑥𝐽 𝐴𝐽))
96, 8imbi12d 343 . . . . . . . 8 (𝑥 = 𝐴 → ((𝑥𝐽 𝑥𝐽) ↔ (𝐴𝐽 𝐴𝐽)))
109spcgv 3585 . . . . . . 7 (𝐴 ∈ 𝒫 𝐽 → (∀𝑥(𝑥𝐽 𝑥𝐽) → (𝐴𝐽 𝐴𝐽)))
115, 10syl 17 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴𝐽) → (∀𝑥(𝑥𝐽 𝑥𝐽) → (𝐴𝐽 𝐴𝐽)))
1211com23 86 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝐽) → (𝐴𝐽 → (∀𝑥(𝑥𝐽 𝑥𝐽) → 𝐴𝐽)))
1312ex 411 . . . 4 (𝐽 ∈ Top → (𝐴𝐽 → (𝐴𝐽 → (∀𝑥(𝑥𝐽 𝑥𝐽) → 𝐴𝐽))))
1413pm2.43d 53 . . 3 (𝐽 ∈ Top → (𝐴𝐽 → (∀𝑥(𝑥𝐽 𝑥𝐽) → 𝐴𝐽)))
153, 14mpid 44 . 2 (𝐽 ∈ Top → (𝐴𝐽 𝐴𝐽))
1615imp 405 1 ((𝐽 ∈ Top ∧ 𝐴𝐽) → 𝐴𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wal 1531   = wceq 1533  wcel 2098  wral 3058  cin 3948  wss 3949  𝒫 cpw 4606   cuni 4912  Topctop 22815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-in 3956  df-ss 3966  df-pw 4608  df-uni 4913  df-top 22816
This theorem is referenced by:  iunopn  22820  unopn  22825  0opn  22826  topopn  22828  tgtop  22896  ntropn  22973  toponmre  23017  neips  23037  txcmplem1  23565  unimopn  24425  metrest  24453  cnopn  24723  locfinreflem  33474  cvmscld  34916  mblfinlem3  37165  mblfinlem4  37166  ismblfin  37167  topclat  48087  toplatlub  48089
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