Theorem iscmp 23417

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 4636	. . 3 ⊢ (𝑥 = 𝐽 → 𝒫 𝑥 = 𝒫 𝐽)
2		unieq 4942	. . . . . 6 ⊢ (𝑥 = 𝐽 → ∪ 𝑥 = ∪ 𝐽)
3		iscmp.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
4	2, 3	eqtr4di 2798	. . . . 5 ⊢ (𝑥 = 𝐽 → ∪ 𝑥 = 𝑋)
5	4	eqeq1d 2742	. . . 4 ⊢ (𝑥 = 𝐽 → (∪ 𝑥 = ∪ 𝑦 ↔ 𝑋 = ∪ 𝑦))
6	4	eqeq1d 2742	. . . . 5 ⊢ (𝑥 = 𝐽 → (∪ 𝑥 = ∪ 𝑧 ↔ 𝑋 = ∪ 𝑧))
7	6	rexbidv 3185	. . . 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 3354	. 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 23416	. 2 ⊢ Comp = {𝑥 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑥(∪ 𝑥 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑥 = ∪ 𝑧)}
11	9, 10	elrab2 3711	1 ⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076   ∩ cin 3975  𝒫 cpw 4622  ∪ cuni 4931  Fincfn 9003  Topctop 22920  Compccmp 23415

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-ss 3993  df-pw 4624  df-uni 4932  df-cmp 23416

This theorem is referenced by:  cmpcov  23418  cncmp  23421  fincmp  23422  cmptop  23424  cmpsub  23429  tgcmp  23430  uncmp  23432  sscmp  23434  cmpfi  23437  comppfsc  23561  txcmp  23672  alexsubb  24075  alexsubALT  24080  cmpcref  33796  onsucsuccmpi  36409  limsucncmpi  36411  pibp16  37379  heibor  37781