Theorem sscmp 23434

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 22940	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2	1	3ad2ant1 1133	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) → 𝐽 ∈ Top)
3		elpwi 4629	. . . 4 ⊢ (𝑥 ∈ 𝒫 𝐽 → 𝑥 ⊆ 𝐽)
4		simpl2 1192	. . . . . . 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 1193	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ (𝑥 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑥)) → 𝐽 ⊆ 𝐾)
7	5, 6	sstrd 4019	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ (𝑥 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑥)) → 𝑥 ⊆ 𝐾)
8		simpl1 1191	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ (𝑥 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑥)) → 𝐽 ∈ (TopOn‘𝑋))
9		toponuni 22941	. . . . . . . . 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 2780	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ (𝑥 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑥)) → 𝑋 = ∪ 𝑥)
13		sscmp.1	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐾
14	13	cmpcov 23418	. . . . . . 7 ⊢ ((𝐾 ∈ Comp ∧ 𝑥 ⊆ 𝐾 ∧ 𝑋 = ∪ 𝑥) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦)
15	4, 7, 12, 14	syl3anc 1371	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ (𝑥 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑥)) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦)
16	10	eqeq1d 2742	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ (𝑥 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑥)) → (𝑋 = ∪ 𝑦 ↔ ∪ 𝐽 = ∪ 𝑦))
17	16	rexbidv 3185	. . . . . 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 592	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ 𝒫 𝐽) → (∪ 𝐽 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)∪ 𝐽 = ∪ 𝑦))
21	20	ralrimiva 3152	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) → ∀𝑥 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)∪ 𝐽 = ∪ 𝑦))
22		eqid 2740	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
23	22	iscmp 23417	. 2 ⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)∪ 𝐽 = ∪ 𝑦)))
24	2, 21, 23	sylanbrc 582	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) → 𝐽 ∈ Comp)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076   ∩ cin 3975   ⊆ wss 3976  𝒫 cpw 4622  ∪ cuni 4931  ‘cfv 6573  Fincfn 9003  Topctop 22920  TopOnctopon 22937  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-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581  df-topon 22938  df-cmp 23416

This theorem is referenced by:  kgencmp2  23575  kgen2ss  23584