Theorem sscmp 23292

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 22800	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2	1	3ad2ant1 1133	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) → 𝐽 ∈ Top)
3		elpwi 4570	. . . 4 ⊢ (𝑥 ∈ 𝒫 𝐽 → 𝑥 ⊆ 𝐽)
4		simpl2 1193	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ (𝑥 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑥)) → 𝐾 ∈ Comp)
5		simprl 770	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ (𝑥 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑥)) → 𝑥 ⊆ 𝐽)
6		simpl3 1194	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ (𝑥 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑥)) → 𝐽 ⊆ 𝐾)
7	5, 6	sstrd 3957	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ (𝑥 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑥)) → 𝑥 ⊆ 𝐾)
8		simpl1 1192	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ (𝑥 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑥)) → 𝐽 ∈ (TopOn‘𝑋))
9		toponuni 22801	. . . . . . . . 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 772	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ (𝑥 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑥)) → ∪ 𝐽 = ∪ 𝑥)
12	10, 11	eqtrd 2764	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ (𝑥 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑥)) → 𝑋 = ∪ 𝑥)
13		sscmp.1	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐾
14	13	cmpcov 23276	. . . . . . 7 ⊢ ((𝐾 ∈ Comp ∧ 𝑥 ⊆ 𝐾 ∧ 𝑋 = ∪ 𝑥) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦)
15	4, 7, 12, 14	syl3anc 1373	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ (𝑥 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑥)) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦)
16	10	eqeq1d 2731	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ (𝑥 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑥)) → (𝑋 = ∪ 𝑦 ↔ ∪ 𝐽 = ∪ 𝑦))
17	16	rexbidv 3157	. . . . . 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 232	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ (𝑥 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑥)) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)∪ 𝐽 = ∪ 𝑦)
19	18	expr 456	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ⊆ 𝐽) → (∪ 𝐽 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)∪ 𝐽 = ∪ 𝑦))
20	3, 19	sylan2 593	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ 𝒫 𝐽) → (∪ 𝐽 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)∪ 𝐽 = ∪ 𝑦))
21	20	ralrimiva 3125	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) → ∀𝑥 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)∪ 𝐽 = ∪ 𝑦))
22		eqid 2729	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
23	22	iscmp 23275	. 2 ⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)∪ 𝐽 = ∪ 𝑦)))
24	2, 21, 23	sylanbrc 583	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) → 𝐽 ∈ Comp)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053   ∩ cin 3913   ⊆ wss 3914  𝒫 cpw 4563  ∪ cuni 4871  ‘cfv 6511  Fincfn 8918  Topctop 22780  TopOnctopon 22797  Compccmp 23273

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-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6464  df-fun 6513  df-fv 6519  df-topon 22798  df-cmp 23274

This theorem is referenced by:  kgencmp2  23433  kgen2ss  23442