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Theorem inopn 20902
Description: The intersection of two open sets of a topology is an open set. (Contributed by NM, 17-Jul-2006.)
Assertion
Ref Expression
inopn ((𝐽 ∈ Top ∧ 𝐴𝐽𝐵𝐽) → (𝐴𝐵) ∈ 𝐽)

Proof of Theorem inopn
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 istopg 20898 . . . . 5 (𝐽 ∈ Top → (𝐽 ∈ Top ↔ (∀𝑥(𝑥𝐽 𝑥𝐽) ∧ ∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽)))
21ibi 256 . . . 4 (𝐽 ∈ Top → (∀𝑥(𝑥𝐽 𝑥𝐽) ∧ ∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽))
32simprd 482 . . 3 (𝐽 ∈ Top → ∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽)
4 ineq1 3946 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑦) = (𝐴𝑦))
54eleq1d 2820 . . . 4 (𝑥 = 𝐴 → ((𝑥𝑦) ∈ 𝐽 ↔ (𝐴𝑦) ∈ 𝐽))
6 ineq2 3947 . . . . 5 (𝑦 = 𝐵 → (𝐴𝑦) = (𝐴𝐵))
76eleq1d 2820 . . . 4 (𝑦 = 𝐵 → ((𝐴𝑦) ∈ 𝐽 ↔ (𝐴𝐵) ∈ 𝐽))
85, 7rspc2v 3457 . . 3 ((𝐴𝐽𝐵𝐽) → (∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽 → (𝐴𝐵) ∈ 𝐽))
93, 8syl5com 31 . 2 (𝐽 ∈ Top → ((𝐴𝐽𝐵𝐽) → (𝐴𝐵) ∈ 𝐽))
1093impib 1109 1 ((𝐽 ∈ Top ∧ 𝐴𝐽𝐵𝐽) → (𝐴𝐵) ∈ 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072  wal 1626   = wceq 1628  wcel 2135  wral 3046  cin 3710  wss 3711   cuni 4584  Topctop 20896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-sep 4929
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ral 3051  df-v 3338  df-in 3718  df-ss 3725  df-pw 4300  df-top 20897
This theorem is referenced by:  fitop  20903  tgclb  20972  topbas  20974  difopn  21036  uncld  21043  ntrin  21063  toponmre  21095  innei  21127  restopnb  21177  ordtopn3  21198  cnprest  21291  islly2  21485  kgentopon  21539  llycmpkgen2  21551  ptbasin  21578  txcnp  21621  txcnmpt  21625  qtoptop2  21700  opnfbas  21843  hauspwpwf1  21988  mopnin  22499  reconnlem2  22827  lmxrge0  30303  cvmsss2  31559  cvmcov2  31560  icccncfext  40599
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