Theorem iscmp 22447

Description: The predicate "is a compact topology". (Contributed by FL, 22-Dec-2008.) (Revised by Mario Carneiro, 11-Feb-2015.)

Hypothesis

Ref	Expression
iscmp.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
iscmp	⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧)))

Distinct variable group:   𝑦,𝑧,𝐽

Allowed substitution hints:   𝑋(𝑦,𝑧)

Proof of Theorem iscmp

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pweq 4546	. . 3 ⊢ (𝑥 = 𝐽 → 𝒫 𝑥 = 𝒫 𝐽)
2		unieq 4847	. . . . . 6 ⊢ (𝑥 = 𝐽 → ∪ 𝑥 = ∪ 𝐽)
3		iscmp.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
4	2, 3	eqtr4di 2797	. . . . 5 ⊢ (𝑥 = 𝐽 → ∪ 𝑥 = 𝑋)
5	4	eqeq1d 2740	. . . 4 ⊢ (𝑥 = 𝐽 → (∪ 𝑥 = ∪ 𝑦 ↔ 𝑋 = ∪ 𝑦))
6	4	eqeq1d 2740	. . . . 5 ⊢ (𝑥 = 𝐽 → (∪ 𝑥 = ∪ 𝑧 ↔ 𝑋 = ∪ 𝑧))
7	6	rexbidv 3225	. . . 4 ⊢ (𝑥 = 𝐽 → (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑥 = ∪ 𝑧 ↔ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧))
8	5, 7	imbi12d 344	. . 3 ⊢ (𝑥 = 𝐽 → ((∪ 𝑥 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑥 = ∪ 𝑧) ↔ (𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧)))
9	1, 8	raleqbidv 3327	. 2 ⊢ (𝑥 = 𝐽 → (∀𝑦 ∈ 𝒫 𝑥(∪ 𝑥 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑥 = ∪ 𝑧) ↔ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧)))
10		df-cmp 22446	. 2 ⊢ Comp = {𝑥 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑥(∪ 𝑥 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑥 = ∪ 𝑧)}
11	9, 10	elrab2 3620	1 ⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064   ∩ cin 3882  𝒫 cpw 4530  ∪ cuni 4836  Fincfn 8691  Topctop 21950  Compccmp 22445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-in 3890  df-ss 3900  df-pw 4532  df-uni 4837  df-cmp 22446

This theorem is referenced by:  cmpcov  22448  cncmp  22451  fincmp  22452  cmptop  22454  cmpsub  22459  tgcmp  22460  uncmp  22462  sscmp  22464  cmpfi  22467  comppfsc  22591  txcmp  22702  alexsubb  23105  alexsubALT  23110  cmpcref  31702  onsucsuccmpi  34559  limsucncmpi  34561  pibp16  35511  heibor  35906