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Theorem inopn 13980
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 13976 . . . . 5 (𝐽 ∈ Top → (𝐽 ∈ Top ↔ (∀𝑥(𝑥𝐽 𝑥𝐽) ∧ ∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽)))
21ibi 176 . . . 4 (𝐽 ∈ Top → (∀𝑥(𝑥𝐽 𝑥𝐽) ∧ ∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽))
32simprd 114 . . 3 (𝐽 ∈ Top → ∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽)
4 ineq1 3344 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑦) = (𝐴𝑦))
54eleq1d 2258 . . . 4 (𝑥 = 𝐴 → ((𝑥𝑦) ∈ 𝐽 ↔ (𝐴𝑦) ∈ 𝐽))
6 ineq2 3345 . . . . 5 (𝑦 = 𝐵 → (𝐴𝑦) = (𝐴𝐵))
76eleq1d 2258 . . . 4 (𝑦 = 𝐵 → ((𝐴𝑦) ∈ 𝐽 ↔ (𝐴𝐵) ∈ 𝐽))
85, 7rspc2v 2869 . . 3 ((𝐴𝐽𝐵𝐽) → (∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽 → (𝐴𝐵) ∈ 𝐽))
93, 8syl5com 29 . 2 (𝐽 ∈ Top → ((𝐴𝐽𝐵𝐽) → (𝐴𝐵) ∈ 𝐽))
1093impib 1203 1 ((𝐽 ∈ Top ∧ 𝐴𝐽𝐵𝐽) → (𝐴𝐵) ∈ 𝐽)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 980  wal 1362   = wceq 1364  wcel 2160  wral 2468  cin 3143  wss 3144   cuni 3824  Topctop 13974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-sep 4136
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-in 3150  df-ss 3157  df-pw 3592  df-top 13975
This theorem is referenced by:  tgclb  14042  topbas  14044  difopn  14085  uncld  14090  ntrin  14101  innei  14140  restopnb  14158  cnptoprest  14216  txcnp  14248  txcnmpt  14250  mopnin  14464
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