Theorem sscmp 23523

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 23031	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2	1	3ad2ant1 1149	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) → 𝐽 ∈ Top)
3		elpwi 4565	. . . 4 ⊢ (𝑥 ∈ 𝒫 𝐽 → 𝑥 ⊆ 𝐽)
4		simpl2 1209	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ (𝑥 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑥)) → 𝐾 ∈ Comp)
5		simprl 782	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ (𝑥 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑥)) → 𝑥 ⊆ 𝐽)
6		simpl3 1210	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ (𝑥 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑥)) → 𝐽 ⊆ 𝐾)
7	5, 6	sstrd 3949	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ (𝑥 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑥)) → 𝑥 ⊆ 𝐾)
8		simpl1 1208	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ (𝑥 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑥)) → 𝐽 ∈ (TopOn‘𝑋))
9		toponuni 23032	. . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
10	8, 9	syl 18	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ (𝑥 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑥)) → 𝑋 = ∪ 𝐽)
11		simprr 784	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ (𝑥 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑥)) → ∪ 𝐽 = ∪ 𝑥)
12	10, 11	eqtrd 2800	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ (𝑥 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑥)) → 𝑋 = ∪ 𝑥)
13		sscmp.1	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐾
14	13	cmpcov 23507	. . . . . . 7 ⊢ ((𝐾 ∈ Comp ∧ 𝑥 ⊆ 𝐾 ∧ 𝑋 = ∪ 𝑥) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦)
15	4, 7, 12, 14	syl3anc 1394	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ (𝑥 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑥)) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦)
16	10	eqeq1d 2767	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ (𝑥 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑥)) → (𝑋 = ∪ 𝑦 ↔ ∪ 𝐽 = ∪ 𝑦))
17	16	rexbidv 3189	. . . . . 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 461	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ⊆ 𝐽) → (∪ 𝐽 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)∪ 𝐽 = ∪ 𝑦))
20	3, 19	sylan2 604	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ 𝒫 𝐽) → (∪ 𝐽 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)∪ 𝐽 = ∪ 𝑦))
21	20	ralrimiva 3157	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) → ∀𝑥 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)∪ 𝐽 = ∪ 𝑦))
22		eqid 2765	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
23	22	iscmp 23506	. 2 ⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)∪ 𝐽 = ∪ 𝑦)))
24	2, 21, 23	sylanbrc 594	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽 ⊆ 𝐾) → 𝐽 ∈ Comp)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  ∀wral 3079  ∃wrex 3089   ∩ cin 3906   ⊆ wss 3907  𝒫 cpw 4558  ∪ cuni 4868  ‘cfv 6525  Fincfn 8931  Topctop 23011  TopOnctopon 23028  Compccmp 23504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-iota 6481  df-fun 6527  df-fv 6533  df-topon 23029  df-cmp 23505

This theorem is referenced by:  kgencmp2  23664  kgen2ss  23673