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Theorem inopn 12184
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 12180 . . . . 5 (𝐽 ∈ Top → (𝐽 ∈ Top ↔ (∀𝑥(𝑥𝐽 𝑥𝐽) ∧ ∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽)))
21ibi 175 . . . 4 (𝐽 ∈ Top → (∀𝑥(𝑥𝐽 𝑥𝐽) ∧ ∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽))
32simprd 113 . . 3 (𝐽 ∈ Top → ∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽)
4 ineq1 3270 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑦) = (𝐴𝑦))
54eleq1d 2208 . . . 4 (𝑥 = 𝐴 → ((𝑥𝑦) ∈ 𝐽 ↔ (𝐴𝑦) ∈ 𝐽))
6 ineq2 3271 . . . . 5 (𝑦 = 𝐵 → (𝐴𝑦) = (𝐴𝐵))
76eleq1d 2208 . . . 4 (𝑦 = 𝐵 → ((𝐴𝑦) ∈ 𝐽 ↔ (𝐴𝐵) ∈ 𝐽))
85, 7rspc2v 2802 . . 3 ((𝐴𝐽𝐵𝐽) → (∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽 → (𝐴𝐵) ∈ 𝐽))
93, 8syl5com 29 . 2 (𝐽 ∈ Top → ((𝐴𝐽𝐵𝐽) → (𝐴𝐵) ∈ 𝐽))
1093impib 1179 1 ((𝐽 ∈ Top ∧ 𝐴𝐽𝐵𝐽) → (𝐴𝐵) ∈ 𝐽)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 962  wal 1329   = wceq 1331  wcel 1480  wral 2416  cin 3070  wss 3071   cuni 3736  Topctop 12178
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-in 3077  df-ss 3084  df-pw 3512  df-top 12179
This theorem is referenced by:  tgclb  12248  topbas  12250  difopn  12291  uncld  12296  ntrin  12307  innei  12346  restopnb  12364  cnptoprest  12422  txcnp  12454  txcnmpt  12456  mopnin  12670
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