Theorem sscmp 21018

Description: A subset of a compact topology (i.e. a coarser topology) is compact. (Contributed by Mario Carneiro, 20-Mar-2015.)

Hypothesis

Ref	Expression
sscmp.1	⊢ 𝑋 = ∪ 𝐾

Assertion

Ref	Expression
sscmp	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) → 𝐽 ∈ Comp)

Proof of Theorem sscmp

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		topontop 20541	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2	1	3ad2ant1 1075	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) → 𝐽 ∈ Top)
3		elpwi 4117	. . . 4 ⊢ (𝑥 ∈ 𝒫 𝐽 → 𝑥 ⊆ 𝐽)
4		simpl2 1058	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ (𝑥 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑥)) → 𝐾 ∈ Comp)
5		simprl 790	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ (𝑥 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑥)) → 𝑥 ⊆ 𝐽)
6		simpl3 1059	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ (𝑥 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑥)) → 𝐽 ⊆ 𝐾)
7	5, 6	sstrd 3578	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ (𝑥 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑥)) → 𝑥 ⊆ 𝐾)
8		simpl1 1057	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ (𝑥 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑥)) → 𝐽 ∈ (TopOn‘𝑋))
9		toponuni 20542	. . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
10	8, 9	syl 17	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ (𝑥 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑥)) → 𝑋 = ∪ 𝐽)
11		simprr 792	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ (𝑥 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑥)) → ∪ 𝐽 = ∪ 𝑥)
12	10, 11	eqtrd 2644	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ (𝑥 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑥)) → 𝑋 = ∪ 𝑥)
13		sscmp.1	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐾
14	13	cmpcov 21002	. . . . . . 7 ⊢ ((𝐾 ∈ Comp ∧ 𝑥 ⊆ 𝐾 ∧ 𝑋 = ∪ 𝑥) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦)
15	4, 7, 12, 14	syl3anc 1318	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ (𝑥 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑥)) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦)
16	10	eqeq1d 2612	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ (𝑥 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑥)) → (𝑋 = ∪ 𝑦 ↔ ∪ 𝐽 = ∪ 𝑦))
17	16	rexbidv 3034	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ (𝑥 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑥)) → (∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦 ↔ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)∪ 𝐽 = ∪ 𝑦))
18	15, 17	mpbid 221	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ (𝑥 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑥)) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)∪ 𝐽 = ∪ 𝑦)
19	18	expr 641	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ⊆ 𝐽) → (∪ 𝐽 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)∪ 𝐽 = ∪ 𝑦))
20	3, 19	sylan2 490	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ 𝒫 𝐽) → (∪ 𝐽 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)∪ 𝐽 = ∪ 𝑦))
21	20	ralrimiva 2949	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) → ∀𝑥 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)∪ 𝐽 = ∪ 𝑦))
22		eqid 2610	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
23	22	iscmp 21001	. 2 ⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)∪ 𝐽 = ∪ 𝑦)))
24	2, 21, 23	sylanbrc 695	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) → 𝐽 ∈ Comp)

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372  ‘cfv 5804  Fincfn 7841  Topctop 20517  TopOnctopon 20518  Compccmp 20999

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-topon 20523  df-cmp 21000

This theorem is referenced by:  kgencmp2  21159  kgen2ss  21168