Theorem inopn 23093

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 23089	. . . . 5 ⊢ (𝐽 ∈ Top → (𝐽 ∈ Top ↔ (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 𝐽)))
2	1	ibi 270	. . . 4 ⊢ (𝐽 ∈ Top → (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 𝐽))
3	2	simprd 501	. . 3 ⊢ (𝐽 ∈ Top → ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 𝐽)
4		ineq1 4169	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∩ 𝑦) = (𝐴 ∩ 𝑦))
5	4	eleq1d 2851	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∩ 𝑦) ∈ 𝐽 ↔ (𝐴 ∩ 𝑦) ∈ 𝐽))
6		ineq2 4170	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ∩ 𝑦) = (𝐴 ∩ 𝐵))
7	6	eleq1d 2851	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ∩ 𝑦) ∈ 𝐽 ↔ (𝐴 ∩ 𝐵) ∈ 𝐽))
8	5, 7	rspc2v 3595	. . 3 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → (∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 𝐽 → (𝐴 ∩ 𝐵) ∈ 𝐽))
9	3, 8	syl5com 32	. 2 ⊢ (𝐽 ∈ Top → ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → (𝐴 ∩ 𝐵) ∈ 𝐽))
10	9	3impib 1134	1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → (𝐴 ∩ 𝐵) ∈ 𝐽)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   = wceq 1570   ∈ wcel 2146  ∀wral 3082   ∩ cin 3907   ⊆ wss 3908  ∪ cuni 4877  Topctop 23087

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-in 3915  df-ss 3925  df-pw 4569  df-top 23088

This theorem is used by:  fitop  23094  tgclb  23164  topbas  23166  difopn  23228  uncld  23235  ntrin  23255  toponmre  23287  innei  23319  restopnb  23369  ordtopn3  23390  cnprest  23483  islly2  23678  kgentopon  23732  llycmpkgen2  23744  ptbasin  23771  txcnp  23814  txcnmpt  23818  qtoptop2  23893  opnfbas  24036  hauspwpwf1  24181  mopnin  24691  reconnlem2  25022  lmxrge0  34373  cvmsss2  35787  cvmcov2  35788  inopnd  45908  icccncfext  46642  toplatmeet  49822  topdlat  49823