Theorem kgen2ss 23442

Description: The compact generator preserves the subset (fineness) relationship on topologies. (Contributed by Mario Carneiro, 21-Mar-2015.)

Assertion

Ref	Expression
kgen2ss	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (𝑘Gen‘𝐽) ⊆ (𝑘Gen‘𝐾))

Proof of Theorem kgen2ss

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

Step	Hyp	Ref	Expression
1		simp1 1136	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → 𝐽 ∈ (TopOn‘𝑋))
2		elpwi 4570	. . . . . . . . 9 ⊢ (𝑘 ∈ 𝒫 𝑋 → 𝑘 ⊆ 𝑋)
3		resttopon 23048	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑘 ⊆ 𝑋) → (𝐽 ↾t 𝑘) ∈ (TopOn‘𝑘))
4	1, 2, 3	syl2an 596	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → (𝐽 ↾t 𝑘) ∈ (TopOn‘𝑘))
5		simp2 1137	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → 𝐾 ∈ (TopOn‘𝑋))
6		resttopon 23048	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝑘 ⊆ 𝑋) → (𝐾 ↾t 𝑘) ∈ (TopOn‘𝑘))
7	5, 2, 6	syl2an 596	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → (𝐾 ↾t 𝑘) ∈ (TopOn‘𝑘))
8		toponuni 22801	. . . . . . . . . 10 ⊢ ((𝐾 ↾t 𝑘) ∈ (TopOn‘𝑘) → 𝑘 = ∪ (𝐾 ↾t 𝑘))
9	7, 8	syl 17	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝑘 = ∪ (𝐾 ↾t 𝑘))
10	9	fveq2d 6862	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → (TopOn‘𝑘) = (TopOn‘∪ (𝐾 ↾t 𝑘)))
11	4, 10	eleqtrd 2830	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → (𝐽 ↾t 𝑘) ∈ (TopOn‘∪ (𝐾 ↾t 𝑘)))
12		simpl2 1193	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝐾 ∈ (TopOn‘𝑋))
13		topontop 22800	. . . . . . . . 9 ⊢ (𝐾 ∈ (TopOn‘𝑋) → 𝐾 ∈ Top)
14	12, 13	syl 17	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝐾 ∈ Top)
15		simpl3 1194	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝐽 ⊆ 𝐾)
16		ssrest 23063	. . . . . . . 8 ⊢ ((𝐾 ∈ Top ∧ 𝐽 ⊆ 𝐾) → (𝐽 ↾t 𝑘) ⊆ (𝐾 ↾t 𝑘))
17	14, 15, 16	syl2anc 584	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → (𝐽 ↾t 𝑘) ⊆ (𝐾 ↾t 𝑘))
18		eqid 2729	. . . . . . . . . 10 ⊢ ∪ (𝐾 ↾t 𝑘) = ∪ (𝐾 ↾t 𝑘)
19	18	sscmp 23292	. . . . . . . . 9 ⊢ (((𝐽 ↾t 𝑘) ∈ (TopOn‘∪ (𝐾 ↾t 𝑘)) ∧ (𝐾 ↾t 𝑘) ∈ Comp ∧ (𝐽 ↾t 𝑘) ⊆ (𝐾 ↾t 𝑘)) → (𝐽 ↾t 𝑘) ∈ Comp)
20	19	3com23 1126	. . . . . . . 8 ⊢ (((𝐽 ↾t 𝑘) ∈ (TopOn‘∪ (𝐾 ↾t 𝑘)) ∧ (𝐽 ↾t 𝑘) ⊆ (𝐾 ↾t 𝑘) ∧ (𝐾 ↾t 𝑘) ∈ Comp) → (𝐽 ↾t 𝑘) ∈ Comp)
21	20	3expia 1121	. . . . . . 7 ⊢ (((𝐽 ↾t 𝑘) ∈ (TopOn‘∪ (𝐾 ↾t 𝑘)) ∧ (𝐽 ↾t 𝑘) ⊆ (𝐾 ↾t 𝑘)) → ((𝐾 ↾t 𝑘) ∈ Comp → (𝐽 ↾t 𝑘) ∈ Comp))
22	11, 17, 21	syl2anc 584	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → ((𝐾 ↾t 𝑘) ∈ Comp → (𝐽 ↾t 𝑘) ∈ Comp))
23	17	sseld 3945	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → ((𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘) → (𝑥 ∩ 𝑘) ∈ (𝐾 ↾t 𝑘)))
24	22, 23	imim12d 81	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → (((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)) → ((𝐾 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐾 ↾t 𝑘))))
25	24	ralimdva 3145	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)) → ∀𝑘 ∈ 𝒫 𝑋((𝐾 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐾 ↾t 𝑘))))
26	25	anim2d 612	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → ((𝑥 ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))) → (𝑥 ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐾 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐾 ↾t 𝑘)))))
27		elkgen 23423	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑥 ∈ (𝑘Gen‘𝐽) ↔ (𝑥 ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)))))
28	27	3ad2ant1 1133	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (𝑥 ∈ (𝑘Gen‘𝐽) ↔ (𝑥 ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)))))
29		elkgen 23423	. . . 4 ⊢ (𝐾 ∈ (TopOn‘𝑋) → (𝑥 ∈ (𝑘Gen‘𝐾) ↔ (𝑥 ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐾 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐾 ↾t 𝑘)))))
30	29	3ad2ant2 1134	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (𝑥 ∈ (𝑘Gen‘𝐾) ↔ (𝑥 ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐾 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐾 ↾t 𝑘)))))
31	26, 28, 30	3imtr4d 294	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (𝑥 ∈ (𝑘Gen‘𝐽) → 𝑥 ∈ (𝑘Gen‘𝐾)))
32	31	ssrdv 3952	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (𝑘Gen‘𝐽) ⊆ (𝑘Gen‘𝐾))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ∩ cin 3913   ⊆ wss 3914  𝒫 cpw 4563  ∪ cuni 4871  ‘cfv 6511  (class class class)co 7387   ↾_t crest 17383  Topctop 22780  TopOnctopon 22797  Compccmp 23273  𝑘Genckgen 23420

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-rep 5234  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-3or 1087  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-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-en 8919  df-fin 8922  df-fi 9362  df-rest 17385  df-topgen 17406  df-top 22781  df-topon 22798  df-bases 22833  df-cmp 23274  df-kgen 23421

This theorem is referenced by: (None)