Theorem inopn 22786

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 22782	. . . . 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 495	. . 3 ⊢ (𝐽 ∈ Top → ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 𝐽)
4		ineq1 4176	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∩ 𝑦) = (𝐴 ∩ 𝑦))
5	4	eleq1d 2813	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∩ 𝑦) ∈ 𝐽 ↔ (𝐴 ∩ 𝑦) ∈ 𝐽))
6		ineq2 4177	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ∩ 𝑦) = (𝐴 ∩ 𝐵))
7	6	eleq1d 2813	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ∩ 𝑦) ∈ 𝐽 ↔ (𝐴 ∩ 𝐵) ∈ 𝐽))
8	5, 7	rspc2v 3599	. . 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 395   ∧ w3a 1086  ∀wal 1538   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ∩ cin 3913   ⊆ wss 3914  ∪ cuni 4871  Topctop 22780

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-in 3921  df-ss 3931  df-pw 4565  df-top 22781

This theorem is referenced by:  fitop  22787  tgclb  22857  topbas  22859  difopn  22921  uncld  22928  ntrin  22948  toponmre  22980  innei  23012  restopnb  23062  ordtopn3  23083  cnprest  23176  islly2  23371  kgentopon  23425  llycmpkgen2  23437  ptbasin  23464  txcnp  23507  txcnmpt  23511  qtoptop2  23586  opnfbas  23729  hauspwpwf1  23874  mopnin  24385  reconnlem2  24716  lmxrge0  33942  cvmsss2  35261  cvmcov2  35262  inopnd  45143  icccncfext  45885  toplatmeet  48991  topdlat  48992