Theorem kgen2cn 23567

Description: A continuous function is also continuous with the domain and codomain replaced by their compact generator topologies. (Contributed by Mario Carneiro, 21-Mar-2015.)

Assertion

Ref	Expression
kgen2cn	⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹 ∈ ((𝑘Gen‘𝐽) Cn (𝑘Gen‘𝐾)))

Proof of Theorem kgen2cn

Step	Hyp	Ref	Expression
1		cntop1 23248	. . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
2		toptopon2 22924	. . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
3	1, 2	sylib 218	. . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ (TopOn‘∪ 𝐽))
4		kgentopon 23546	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) → (𝑘Gen‘𝐽) ∈ (TopOn‘∪ 𝐽))
5	3, 4	syl 17	. . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → (𝑘Gen‘𝐽) ∈ (TopOn‘∪ 𝐽))
6		kgenss 23551	. . . . 5 ⊢ (𝐽 ∈ Top → 𝐽 ⊆ (𝑘Gen‘𝐽))
7	1, 6	syl 17	. . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ⊆ (𝑘Gen‘𝐽))
8		eqid 2737	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
9	8	cnss1 23284	. . . 4 ⊢ (((𝑘Gen‘𝐽) ∈ (TopOn‘∪ 𝐽) ∧ 𝐽 ⊆ (𝑘Gen‘𝐽)) → (𝐽 Cn 𝐾) ⊆ ((𝑘Gen‘𝐽) Cn 𝐾))
10	5, 7, 9	syl2anc 584	. . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐽 Cn 𝐾) ⊆ ((𝑘Gen‘𝐽) Cn 𝐾))
11		kgenf 23549	. . . . . 6 ⊢ 𝑘Gen:Top⟶Top
12		ffn 6736	. . . . . 6 ⊢ (𝑘Gen:Top⟶Top → 𝑘Gen Fn Top)
13	11, 12	ax-mp 5	. . . . 5 ⊢ 𝑘Gen Fn Top
14		fnfvelrn 7100	. . . . 5 ⊢ ((𝑘Gen Fn Top ∧ 𝐽 ∈ Top) → (𝑘Gen‘𝐽) ∈ ran 𝑘Gen)
15	13, 1, 14	sylancr 587	. . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → (𝑘Gen‘𝐽) ∈ ran 𝑘Gen)
16		cntop2 23249	. . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
17		kgencn3 23566	. . . 4 ⊢ (((𝑘Gen‘𝐽) ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → ((𝑘Gen‘𝐽) Cn 𝐾) = ((𝑘Gen‘𝐽) Cn (𝑘Gen‘𝐾)))
18	15, 16, 17	syl2anc 584	. . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → ((𝑘Gen‘𝐽) Cn 𝐾) = ((𝑘Gen‘𝐽) Cn (𝑘Gen‘𝐾)))
19	10, 18	sseqtrd 4020	. 2 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐽 Cn 𝐾) ⊆ ((𝑘Gen‘𝐽) Cn (𝑘Gen‘𝐾)))
20		id 22	. 2 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
21	19, 20	sseldd 3984	1 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹 ∈ ((𝑘Gen‘𝐽) Cn (𝑘Gen‘𝐾)))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108   ⊆ wss 3951  ∪ cuni 4907  ran crn 5686   Fn wfn 6556  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431  Topctop 22899  TopOnctopon 22916   Cn ccn 23232  𝑘Genckgen 23541

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-1o 8506  df-map 8868  df-en 8986  df-dom 8987  df-fin 8989  df-fi 9451  df-rest 17467  df-topgen 17488  df-top 22900  df-topon 22917  df-bases 22953  df-cn 23235  df-cmp 23395  df-kgen 23542

This theorem is referenced by: (None)