Theorem kgen2ss 23511

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 1137	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → 𝐽 ∈ (TopOn‘𝑋))
2		elpwi 4563	. . . . . . . . 9 ⊢ (𝑘 ∈ 𝒫 𝑋 → 𝑘 ⊆ 𝑋)
3		resttopon 23117	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑘 ⊆ 𝑋) → (𝐽 ↾t 𝑘) ∈ (TopOn‘𝑘))
4	1, 2, 3	syl2an 597	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → (𝐽 ↾t 𝑘) ∈ (TopOn‘𝑘))
5		simp2 1138	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → 𝐾 ∈ (TopOn‘𝑋))
6		resttopon 23117	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝑘 ⊆ 𝑋) → (𝐾 ↾t 𝑘) ∈ (TopOn‘𝑘))
7	5, 2, 6	syl2an 597	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → (𝐾 ↾t 𝑘) ∈ (TopOn‘𝑘))
8		toponuni 22870	. . . . . . . . . 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 6846	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → (TopOn‘𝑘) = (TopOn‘∪ (𝐾 ↾t 𝑘)))
11	4, 10	eleqtrd 2839	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → (𝐽 ↾t 𝑘) ∈ (TopOn‘∪ (𝐾 ↾t 𝑘)))
12		simpl2 1194	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝐾 ∈ (TopOn‘𝑋))
13		topontop 22869	. . . . . . . . 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 1195	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝐽 ⊆ 𝐾)
16		ssrest 23132	. . . . . . . 8 ⊢ ((𝐾 ∈ Top ∧ 𝐽 ⊆ 𝐾) → (𝐽 ↾t 𝑘) ⊆ (𝐾 ↾t 𝑘))
17	14, 15, 16	syl2anc 585	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → (𝐽 ↾t 𝑘) ⊆ (𝐾 ↾t 𝑘))
18		eqid 2737	. . . . . . . . . 10 ⊢ ∪ (𝐾 ↾t 𝑘) = ∪ (𝐾 ↾t 𝑘)
19	18	sscmp 23361	. . . . . . . . 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 1127	. . . . . . . 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 1122	. . . . . . 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 585	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → ((𝐾 ↾t 𝑘) ∈ Comp → (𝐽 ↾t 𝑘) ∈ Comp))
23	17	sseld 3934	. . . . . 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 3150	. . . 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 613	. . 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 23492	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑥 ∈ (𝑘Gen‘𝐽) ↔ (𝑥 ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)))))
28	27	3ad2ant1 1134	. . 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 23492	. . . 4 ⊢ (𝐾 ∈ (TopOn‘𝑋) → (𝑥 ∈ (𝑘Gen‘𝐾) ↔ (𝑥 ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐾 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐾 ↾t 𝑘)))))
30	29	3ad2ant2 1135	. . 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 3941	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (𝑘Gen‘𝐽) ⊆ (𝑘Gen‘𝐾))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ∩ cin 3902   ⊆ wss 3903  𝒫 cpw 4556  ∪ cuni 4865  ‘cfv 6500  (class class class)co 7368   ↾_t crest 17352  Topctop 22849  TopOnctopon 22866  Compccmp 23342  𝑘Genckgen 23489

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-en 8896  df-fin 8899  df-fi 9326  df-rest 17354  df-topgen 17375  df-top 22850  df-topon 22867  df-bases 22902  df-cmp 23343  df-kgen 23490

This theorem is referenced by: (None)