Theorem kgen2cn 22700

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 22381	. . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
2		toptopon2 22057	. . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
3	1, 2	sylib 217	. . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ (TopOn‘∪ 𝐽))
4		kgentopon 22679	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) → (𝑘Gen‘𝐽) ∈ (TopOn‘∪ 𝐽))
5	3, 4	syl 17	. . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → (𝑘Gen‘𝐽) ∈ (TopOn‘∪ 𝐽))
6		kgenss 22684	. . . . 5 ⊢ (𝐽 ∈ Top → 𝐽 ⊆ (𝑘Gen‘𝐽))
7	1, 6	syl 17	. . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ⊆ (𝑘Gen‘𝐽))
8		eqid 2740	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
9	8	cnss1 22417	. . . 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 22682	. . . . . 6 ⊢ 𝑘Gen:Top⟶Top
12		ffn 6597	. . . . . 6 ⊢ (𝑘Gen:Top⟶Top → 𝑘Gen Fn Top)
13	11, 12	ax-mp 5	. . . . 5 ⊢ 𝑘Gen Fn Top
14		fnfvelrn 6953	. . . . 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 22382	. . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
17		kgencn3 22699	. . . 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 3966	. 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 3927	1 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹 ∈ ((𝑘Gen‘𝐽) Cn (𝑘Gen‘𝐾)))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2110   ⊆ wss 3892  ∪ cuni 4845  ran crn 5590   Fn wfn 6426  ⟶wf 6427  ‘cfv 6431  (class class class)co 7269  Topctop 22032  TopOnctopon 22049   Cn ccn 22365  𝑘Genckgen 22674

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 7580

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-tp 4572  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 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6267  df-on 6268  df-lim 6269  df-suc 6270  df-iota 6389  df-fun 6433  df-fn 6434  df-f 6435  df-f1 6436  df-fo 6437  df-f1o 6438  df-fv 6439  df-ov 7272  df-oprab 7273  df-mpo 7274  df-om 7702  df-1st 7818  df-2nd 7819  df-1o 8282  df-er 8473  df-map 8592  df-en 8709  df-dom 8710  df-fin 8712  df-fi 9140  df-rest 17123  df-topgen 17144  df-top 22033  df-topon 22050  df-bases 22086  df-cn 22368  df-cmp 22528  df-kgen 22675

This theorem is referenced by: (None)