Theorem kgen2ss 23867

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 1154	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → 𝐽 ∈ (TopOn‘𝑋))
2		elpwi 4564	. . . . . . . . 9 ⊢ (𝑘 ∈ 𝒫 𝑋 → 𝑘 ⊆ 𝑋)
3		resttopon 23472	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑘 ⊆ 𝑋) → (𝐽 ↾t 𝑘) ∈ (TopOn‘𝑘))
4	1, 2, 3	syl2an 608	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → (𝐽 ↾t 𝑘) ∈ (TopOn‘𝑘))
5		simp2 1155	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → 𝐾 ∈ (TopOn‘𝑋))
6		resttopon 23472	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝑘 ⊆ 𝑋) → (𝐾 ↾t 𝑘) ∈ (TopOn‘𝑘))
7	5, 2, 6	syl2an 608	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → (𝐾 ↾t 𝑘) ∈ (TopOn‘𝑘))
8		toponuni 23225	. . . . . . . . . 10 ⊢ ((𝐾 ↾t 𝑘) ∈ (TopOn‘𝑘) → 𝑘 = ∪ (𝐾 ↾t 𝑘))
9	7, 8	syl 18	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝑘 = ∪ (𝐾 ↾t 𝑘))
10	9	fveq2d 6887	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → (TopOn‘𝑘) = (TopOn‘∪ (𝐾 ↾t 𝑘)))
11	4, 10	eleqtrd 2863	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → (𝐽 ↾t 𝑘) ∈ (TopOn‘∪ (𝐾 ↾t 𝑘)))
12		simpl2 1211	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝐾 ∈ (TopOn‘𝑋))
13		topontop 23224	. . . . . . . . 9 ⊢ (𝐾 ∈ (TopOn‘𝑋) → 𝐾 ∈ Top)
14	12, 13	syl 18	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝐾 ∈ Top)
15		simpl3 1212	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝐽 ⊆ 𝐾)
16		ssrest 23487	. . . . . . . 8 ⊢ ((𝐾 ∈ Top ∧ 𝐽 ⊆ 𝐾) → (𝐽 ↾t 𝑘) ⊆ (𝐾 ↾t 𝑘))
17	14, 15, 16	syl2anc 596	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → (𝐽 ↾t 𝑘) ⊆ (𝐾 ↾t 𝑘))
18		eqid 2761	. . . . . . . . . 10 ⊢ ∪ (𝐾 ↾t 𝑘) = ∪ (𝐾 ↾t 𝑘)
19	18	sscmp 23716	. . . . . . . . 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 1144	. . . . . . . 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 1139	. . . . . . 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 596	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑘 ∈ 𝒫 𝑋) → ((𝐾 ↾t 𝑘) ∈ Comp → (𝐽 ↾t 𝑘) ∈ Comp))
23	17	sseld 3930	. . . . . 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 82	. . . . 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 3175	. . . 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 624	. . 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 23848	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑥 ∈ (𝑘Gen‘𝐽) ↔ (𝑥 ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)))))
28	27	3ad2ant1 1151	. . 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 23848	. . . 4 ⊢ (𝐾 ∈ (TopOn‘𝑋) → (𝑥 ∈ (𝑘Gen‘𝐾) ↔ (𝑥 ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐾 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐾 ↾t 𝑘)))))
30	29	3ad2ant2 1152	. . 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 297	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (𝑥 ∈ (𝑘Gen‘𝐽) → 𝑥 ∈ (𝑘Gen‘𝐾)))
32	31	ssrdv 3937	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (𝑘Gen‘𝐽) ⊆ (𝑘Gen‘𝐾))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3077   ∩ cin 3898   ⊆ wss 3899  𝒫 cpw 4557  ∪ cuni 4867  ‘cfv 6537  (class class class)co 7418   ↾_t crest 17584  Topctop 23204  TopOnctopon 23221  Compccmp 23697  𝑘Genckgen 23845

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-1st 7999  df-2nd 8000  df-en 8967  df-fin 8970  df-fi 9396  df-rest 17586  df-topgen 17607  df-top 23205  df-topon 23222  df-bases 23257  df-cmp 23698  df-kgen 23846

This theorem is used by: (None)