Theorem kgen2cn 23494

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 23175	. . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
2		toptopon2 22853	. . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
3	1, 2	sylib 218	. . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ (TopOn‘∪ 𝐽))
4		kgentopon 23473	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) → (𝑘Gen‘𝐽) ∈ (TopOn‘∪ 𝐽))
5	3, 4	syl 17	. . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → (𝑘Gen‘𝐽) ∈ (TopOn‘∪ 𝐽))
6		kgenss 23478	. . . . 5 ⊢ (𝐽 ∈ Top → 𝐽 ⊆ (𝑘Gen‘𝐽))
7	1, 6	syl 17	. . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ⊆ (𝑘Gen‘𝐽))
8		eqid 2733	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
9	8	cnss1 23211	. . . 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 23476	. . . . . 6 ⊢ 𝑘Gen:Top⟶Top
12		ffn 6659	. . . . . 6 ⊢ (𝑘Gen:Top⟶Top → 𝑘Gen Fn Top)
13	11, 12	ax-mp 5	. . . . 5 ⊢ 𝑘Gen Fn Top
14		fnfvelrn 7022	. . . . 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 23176	. . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
17		kgencn3 23493	. . . 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 3967	. 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 3931	1 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹 ∈ ((𝑘Gen‘𝐽) Cn (𝑘Gen‘𝐾)))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113   ⊆ wss 3898  ∪ cuni 4860  ran crn 5622   Fn wfn 6484  ⟶wf 6485  ‘cfv 6489  (class class class)co 7355  Topctop 22828  TopOnctopon 22845   Cn ccn 23159  𝑘Genckgen 23468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-int 4900  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7358  df-oprab 7359  df-mpo 7360  df-om 7806  df-1st 7930  df-2nd 7931  df-1o 8394  df-map 8761  df-en 8880  df-dom 8881  df-fin 8883  df-fi 9306  df-rest 17333  df-topgen 17354  df-top 22829  df-topon 22846  df-bases 22881  df-cn 23162  df-cmp 23322  df-kgen 23469

This theorem is referenced by: (None)