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Theorem uniopn 11763
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 11761 . . . . 5 (𝐽 ∈ Top → (𝐽 ∈ Top ↔ (∀𝑥(𝑥𝐽 𝑥𝐽) ∧ ∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽)))
21ibi 175 . . . 4 (𝐽 ∈ Top → (∀𝑥(𝑥𝐽 𝑥𝐽) ∧ ∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽))
32simpld 111 . . 3 (𝐽 ∈ Top → ∀𝑥(𝑥𝐽 𝑥𝐽))
4 elpw2g 4000 . . . . . . . 8 (𝐽 ∈ Top → (𝐴 ∈ 𝒫 𝐽𝐴𝐽))
54biimpar 292 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐴𝐽) → 𝐴 ∈ 𝒫 𝐽)
6 sseq1 3050 . . . . . . . . 9 (𝑥 = 𝐴 → (𝑥𝐽𝐴𝐽))
7 unieq 3670 . . . . . . . . . 10 (𝑥 = 𝐴 𝑥 = 𝐴)
87eleq1d 2157 . . . . . . . . 9 (𝑥 = 𝐴 → ( 𝑥𝐽 𝐴𝐽))
96, 8imbi12d 233 . . . . . . . 8 (𝑥 = 𝐴 → ((𝑥𝐽 𝑥𝐽) ↔ (𝐴𝐽 𝐴𝐽)))
109spcgv 2709 . . . . . . 7 (𝐴 ∈ 𝒫 𝐽 → (∀𝑥(𝑥𝐽 𝑥𝐽) → (𝐴𝐽 𝐴𝐽)))
115, 10syl 14 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴𝐽) → (∀𝑥(𝑥𝐽 𝑥𝐽) → (𝐴𝐽 𝐴𝐽)))
1211com23 78 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝐽) → (𝐴𝐽 → (∀𝑥(𝑥𝐽 𝑥𝐽) → 𝐴𝐽)))
1312ex 114 . . . 4 (𝐽 ∈ Top → (𝐴𝐽 → (𝐴𝐽 → (∀𝑥(𝑥𝐽 𝑥𝐽) → 𝐴𝐽))))
1413pm2.43d 50 . . 3 (𝐽 ∈ Top → (𝐴𝐽 → (∀𝑥(𝑥𝐽 𝑥𝐽) → 𝐴𝐽)))
153, 14mpid 42 . 2 (𝐽 ∈ Top → (𝐴𝐽 𝐴𝐽))
1615imp 123 1 ((𝐽 ∈ Top ∧ 𝐴𝐽) → 𝐴𝐽)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wal 1288   = wceq 1290  wcel 1439  wral 2360  cin 3001  wss 3002  𝒫 cpw 3435   cuni 3661  Topctop 11759
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3965
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2624  df-in 3008  df-ss 3015  df-pw 3437  df-uni 3662  df-top 11760
This theorem is referenced by:  iunopn  11764  unopn  11767  0opn  11768  topopn  11770  tgtop  11831  ntropn  11880  neipsm  11917
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