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Theorem inopn 20636
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 20632 . . . . 5 (𝐽 ∈ Top → (𝐽 ∈ Top ↔ (∀𝑥(𝑥𝐽 𝑥𝐽) ∧ ∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽)))
21ibi 256 . . . 4 (𝐽 ∈ Top → (∀𝑥(𝑥𝐽 𝑥𝐽) ∧ ∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽))
32simprd 479 . . 3 (𝐽 ∈ Top → ∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽)
4 ineq1 3790 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑦) = (𝐴𝑦))
54eleq1d 2683 . . . 4 (𝑥 = 𝐴 → ((𝑥𝑦) ∈ 𝐽 ↔ (𝐴𝑦) ∈ 𝐽))
6 ineq2 3791 . . . . 5 (𝑦 = 𝐵 → (𝐴𝑦) = (𝐴𝐵))
76eleq1d 2683 . . . 4 (𝑦 = 𝐵 → ((𝐴𝑦) ∈ 𝐽 ↔ (𝐴𝐵) ∈ 𝐽))
85, 7rspc2v 3310 . . 3 ((𝐴𝐽𝐵𝐽) → (∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽 → (𝐴𝐵) ∈ 𝐽))
93, 8syl5com 31 . 2 (𝐽 ∈ Top → ((𝐴𝐽𝐵𝐽) → (𝐴𝐵) ∈ 𝐽))
1093impib 1259 1 ((𝐽 ∈ Top ∧ 𝐴𝐽𝐵𝐽) → (𝐴𝐵) ∈ 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036  wal 1478   = wceq 1480  wcel 1987  wral 2907  cin 3558  wss 3559   cuni 4407  Topctop 20630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-v 3191  df-in 3566  df-ss 3573  df-pw 4137  df-top 20631
This theorem is referenced by:  fitop  20637  tgclb  20698  topbas  20700  difopn  20761  uncld  20768  ntrin  20788  toponmre  20820  innei  20852  restopnb  20902  ordtopn3  20923  cnprest  21016  islly2  21210  kgentopon  21264  llycmpkgen2  21276  ptbasin  21303  txcnp  21346  txcnmpt  21350  qtoptop2  21425  opnfbas  21569  hauspwpwf1  21714  mopnin  22225  reconnlem2  22553  lmxrge0  29804  cvmsss2  30999  cvmcov2  31000  icccncfext  39431
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