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Theorem inopn 22905
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 22901 . . . . 5 (𝐽 ∈ Top → (𝐽 ∈ Top ↔ (∀𝑥(𝑥𝐽 𝑥𝐽) ∧ ∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽)))
21ibi 267 . . . 4 (𝐽 ∈ Top → (∀𝑥(𝑥𝐽 𝑥𝐽) ∧ ∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽))
32simprd 495 . . 3 (𝐽 ∈ Top → ∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽)
4 ineq1 4213 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑦) = (𝐴𝑦))
54eleq1d 2826 . . . 4 (𝑥 = 𝐴 → ((𝑥𝑦) ∈ 𝐽 ↔ (𝐴𝑦) ∈ 𝐽))
6 ineq2 4214 . . . . 5 (𝑦 = 𝐵 → (𝐴𝑦) = (𝐴𝐵))
76eleq1d 2826 . . . 4 (𝑦 = 𝐵 → ((𝐴𝑦) ∈ 𝐽 ↔ (𝐴𝐵) ∈ 𝐽))
85, 7rspc2v 3633 . . 3 ((𝐴𝐽𝐵𝐽) → (∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽 → (𝐴𝐵) ∈ 𝐽))
93, 8syl5com 31 . 2 (𝐽 ∈ Top → ((𝐴𝐽𝐵𝐽) → (𝐴𝐵) ∈ 𝐽))
1093impib 1117 1 ((𝐽 ∈ Top ∧ 𝐴𝐽𝐵𝐽) → (𝐴𝐵) ∈ 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087  wal 1538   = wceq 1540  wcel 2108  wral 3061  cin 3950  wss 3951   cuni 4907  Topctop 22899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-in 3958  df-ss 3968  df-pw 4602  df-top 22900
This theorem is referenced by:  fitop  22906  tgclb  22977  topbas  22979  difopn  23042  uncld  23049  ntrin  23069  toponmre  23101  innei  23133  restopnb  23183  ordtopn3  23204  cnprest  23297  islly2  23492  kgentopon  23546  llycmpkgen2  23558  ptbasin  23585  txcnp  23628  txcnmpt  23632  qtoptop2  23707  opnfbas  23850  hauspwpwf1  23995  mopnin  24510  reconnlem2  24849  lmxrge0  33951  cvmsss2  35279  cvmcov2  35280  inopnd  45154  icccncfext  45902  toplatmeet  48892  topdlat  48893
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