Theorem sscmp 22010

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 21518	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2	1	3ad2ant1 1130	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) → 𝐽 ∈ Top)
3		elpwi 4506	. . . 4 ⊢ (𝑥 ∈ 𝒫 𝐽 → 𝑥 ⊆ 𝐽)
4		simpl2 1189	. . . . . . 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 1190	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ (𝑥 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑥)) → 𝐽 ⊆ 𝐾)
7	5, 6	sstrd 3925	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ (𝑥 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑥)) → 𝑥 ⊆ 𝐾)
8		simpl1 1188	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ (𝑥 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑥)) → 𝐽 ∈ (TopOn‘𝑋))
9		toponuni 21519	. . . . . . . . 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 2833	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ (𝑥 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑥)) → 𝑋 = ∪ 𝑥)
13		sscmp.1	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐾
14	13	cmpcov 21994	. . . . . . 7 ⊢ ((𝐾 ∈ Comp ∧ 𝑥 ⊆ 𝐾 ∧ 𝑋 = ∪ 𝑥) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦)
15	4, 7, 12, 14	syl3anc 1368	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ (𝑥 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑥)) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦)
16	10	eqeq1d 2800	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ (𝑥 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑥)) → (𝑋 = ∪ 𝑦 ↔ ∪ 𝐽 = ∪ 𝑦))
17	16	rexbidv 3256	. . . . . 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 235	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ (𝑥 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑥)) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)∪ 𝐽 = ∪ 𝑦)
19	18	expr 460	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ⊆ 𝐽) → (∪ 𝐽 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)∪ 𝐽 = ∪ 𝑦))
20	3, 19	sylan2 595	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ 𝒫 𝐽) → (∪ 𝐽 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)∪ 𝐽 = ∪ 𝑦))
21	20	ralrimiva 3149	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) → ∀𝑥 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)∪ 𝐽 = ∪ 𝑦))
22		eqid 2798	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
23	22	iscmp 21993	. 2 ⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)∪ 𝐽 = ∪ 𝑦)))
24	2, 21, 23	sylanbrc 586	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) → 𝐽 ∈ Comp)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107   ∩ cin 3880   ⊆ wss 3881  𝒫 cpw 4497  ∪ cuni 4800  ‘cfv 6324  Fincfn 8492  Topctop 21498  TopOnctopon 21515  Compccmp 21991

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-topon 21516  df-cmp 21992

This theorem is referenced by:  kgencmp2  22151  kgen2ss  22160