Theorem cmpkgen 22700

Description: A compact space is compactly generated. (Contributed by Mario Carneiro, 21-Mar-2015.)

Assertion

Ref	Expression
cmpkgen	⊢ (𝐽 ∈ Comp → 𝐽 ∈ ran 𝑘Gen)

Proof of Theorem cmpkgen

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

Step	Hyp	Ref	Expression
1		eqid 2740	. 2 ⊢ ∪ 𝐽 = ∪ 𝐽
2		cmptop 22544	. 2 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top)
3	2	adantr 481	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝑥 ∈ ∪ 𝐽) → 𝐽 ∈ Top)
4	1	topopn 22053	. . . . 5 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ 𝐽)
5	3, 4	syl 17	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝑥 ∈ ∪ 𝐽) → ∪ 𝐽 ∈ 𝐽)
6		simpr 485	. . . . 5 ⊢ ((𝐽 ∈ Comp ∧ 𝑥 ∈ ∪ 𝐽) → 𝑥 ∈ ∪ 𝐽)
7	6	snssd 4748	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝑥 ∈ ∪ 𝐽) → {𝑥} ⊆ ∪ 𝐽)
8		opnneiss 22267	. . . 4 ⊢ ((𝐽 ∈ Top ∧ ∪ 𝐽 ∈ 𝐽 ∧ {𝑥} ⊆ ∪ 𝐽) → ∪ 𝐽 ∈ ((nei‘𝐽)‘{𝑥}))
9	3, 5, 7, 8	syl3anc 1370	. . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝑥 ∈ ∪ 𝐽) → ∪ 𝐽 ∈ ((nei‘𝐽)‘{𝑥}))
10	1	restid 17142	. . . . 5 ⊢ (𝐽 ∈ Top → (𝐽 ↾t ∪ 𝐽) = 𝐽)
11	3, 10	syl 17	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝑥 ∈ ∪ 𝐽) → (𝐽 ↾t ∪ 𝐽) = 𝐽)
12		simpl 483	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝑥 ∈ ∪ 𝐽) → 𝐽 ∈ Comp)
13	11, 12	eqeltrd 2841	. . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝑥 ∈ ∪ 𝐽) → (𝐽 ↾t ∪ 𝐽) ∈ Comp)
14		oveq2 7279	. . . . 5 ⊢ (𝑘 = ∪ 𝐽 → (𝐽 ↾t 𝑘) = (𝐽 ↾t ∪ 𝐽))
15	14	eleq1d 2825	. . . 4 ⊢ (𝑘 = ∪ 𝐽 → ((𝐽 ↾t 𝑘) ∈ Comp ↔ (𝐽 ↾t ∪ 𝐽) ∈ Comp))
16	15	rspcev 3561	. . 3 ⊢ ((∪ 𝐽 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t ∪ 𝐽) ∈ Comp) → ∃𝑘 ∈ ((nei‘𝐽)‘{𝑥})(𝐽 ↾t 𝑘) ∈ Comp)
17	9, 13, 16	syl2anc 584	. 2 ⊢ ((𝐽 ∈ Comp ∧ 𝑥 ∈ ∪ 𝐽) → ∃𝑘 ∈ ((nei‘𝐽)‘{𝑥})(𝐽 ↾t 𝑘) ∈ Comp)
18	1, 2, 17	llycmpkgen2 22699	1 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ ran 𝑘Gen)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1542   ∈ wcel 2110  ∃wrex 3067   ⊆ wss 3892  {csn 4567  ∪ cuni 4845  ran crn 5591  ‘cfv 6432  (class class class)co 7271   ↾_t crest 17129  Topctop 22040  neicnei 22246  Compccmp 22535  𝑘Genckgen 22682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-int 4886  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-ord 6268  df-on 6269  df-lim 6270  df-suc 6271  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-ov 7274  df-oprab 7275  df-mpo 7276  df-om 7707  df-1st 7824  df-2nd 7825  df-en 8717  df-fin 8720  df-fi 9148  df-rest 17131  df-topgen 17152  df-top 22041  df-topon 22058  df-bases 22094  df-ntr 22169  df-nei 22247  df-cmp 22536  df-kgen 22683

This theorem is referenced by: (None)