Theorem inopn 22097

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 ∧ 𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → (𝐴 ∩ 𝐵) ∈ 𝐽)

Proof of Theorem inopn

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		istopg 22093	. . . . 5 ⊢ (𝐽 ∈ Top → (𝐽 ∈ Top ↔ (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 𝐽)))
2	1	ibi 267	. . . 4 ⊢ (𝐽 ∈ Top → (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 𝐽))
3	2	simprd 497	. . 3 ⊢ (𝐽 ∈ Top → ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 𝐽)
4		ineq1 4145	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∩ 𝑦) = (𝐴 ∩ 𝑦))
5	4	eleq1d 2821	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∩ 𝑦) ∈ 𝐽 ↔ (𝐴 ∩ 𝑦) ∈ 𝐽))
6		ineq2 4146	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ∩ 𝑦) = (𝐴 ∩ 𝐵))
7	6	eleq1d 2821	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ∩ 𝑦) ∈ 𝐽 ↔ (𝐴 ∩ 𝐵) ∈ 𝐽))
8	5, 7	rspc2v 3575	. . 3 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → (∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 𝐽 → (𝐴 ∩ 𝐵) ∈ 𝐽))
9	3, 8	syl5com 31	. 2 ⊢ (𝐽 ∈ Top → ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → (𝐴 ∩ 𝐵) ∈ 𝐽))
10	9	3impib 1116	1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → (𝐴 ∩ 𝐵) ∈ 𝐽)

Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1087  ∀wal 1537   = wceq 1539   ∈ wcel 2104  ∀wral 3062   ∩ cin 3891   ⊆ wss 3892  ∪ cuni 4844  Topctop 22091

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707  ax-sep 5232

This theorem depends on definitions:  df-bi 206  df-an 398  df-3an 1089  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3063  df-rab 3306  df-v 3439  df-in 3899  df-ss 3909  df-pw 4541  df-top 22092

This theorem is referenced by:  fitop  22098  tgclb  22169  topbas  22171  difopn  22234  uncld  22241  ntrin  22261  toponmre  22293  innei  22325  restopnb  22375  ordtopn3  22396  cnprest  22489  islly2  22684  kgentopon  22738  llycmpkgen2  22750  ptbasin  22777  txcnp  22820  txcnmpt  22824  qtoptop2  22899  opnfbas  23042  hauspwpwf1  23187  mopnin  23702  reconnlem2  24039  lmxrge0  31951  cvmsss2  33285  cvmcov2  33286  inopnd  42917  icccncfext  43657  toplatmeet  46533  topdlat  46534