Theorem cmptop 21400

Description: A compact topology is a topology. (Contributed by Jeff Hankins, 29-Jun-2009.)

Assertion

Ref	Expression
cmptop	⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top)

Proof of Theorem cmptop

Dummy variables 𝑠 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2760	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	iscmp 21393	. 2 ⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑟 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)∪ 𝐽 = ∪ 𝑠)))
3	2	simplbi 478	1 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top)

Syntax hints:   → wi 4   = wceq 1632   ∈ wcel 2139  ∀wral 3050  ∃wrex 3051   ∩ cin 3714  𝒫 cpw 4302  ∪ cuni 4588  Fincfn 8121  Topctop 20900  Compccmp 21391

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-in 3722  df-ss 3729  df-pw 4304  df-uni 4589  df-cmp 21392

This theorem is referenced by:  imacmp  21402  cmpcld  21407  fiuncmp  21409  cmpfii  21414  bwth  21415  locfincmp  21531  kgeni  21542  kgentopon  21543  kgencmp  21550  kgencmp2  21551  cmpkgen  21556  txcmplem1  21646  txcmp  21648  qtopcmp  21713  cmphaushmeo  21805  ptcmpfi  21818  fclscmpi  22034  alexsubALTlem1  22052  ptcmplem1  22057  ptcmpg  22062  evth  22959  evth2  22960  cmppcmp  30234  ordcmp  32752  poimirlem30  33752  heibor1lem  33921  cmpfiiin  37762  kelac1  38135  kelac2  38137  stoweidlem28  40748  stoweidlem50  40770  stoweidlem53  40773  stoweidlem57  40777  stoweidlem62  40782