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Theorem kgen2cn 22169
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
StepHypRef Expression
1 cntop1 21850 . . . . . 6 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
2 toptopon2 21528 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
31, 2sylib 220 . . . . 5 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ (TopOn‘ 𝐽))
4 kgentopon 22148 . . . . 5 (𝐽 ∈ (TopOn‘ 𝐽) → (𝑘Gen‘𝐽) ∈ (TopOn‘ 𝐽))
53, 4syl 17 . . . 4 (𝐹 ∈ (𝐽 Cn 𝐾) → (𝑘Gen‘𝐽) ∈ (TopOn‘ 𝐽))
6 kgenss 22153 . . . . 5 (𝐽 ∈ Top → 𝐽 ⊆ (𝑘Gen‘𝐽))
71, 6syl 17 . . . 4 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ⊆ (𝑘Gen‘𝐽))
8 eqid 2823 . . . . 5 𝐽 = 𝐽
98cnss1 21886 . . . 4 (((𝑘Gen‘𝐽) ∈ (TopOn‘ 𝐽) ∧ 𝐽 ⊆ (𝑘Gen‘𝐽)) → (𝐽 Cn 𝐾) ⊆ ((𝑘Gen‘𝐽) Cn 𝐾))
105, 7, 9syl2anc 586 . . 3 (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐽 Cn 𝐾) ⊆ ((𝑘Gen‘𝐽) Cn 𝐾))
11 kgenf 22151 . . . . . 6 𝑘Gen:Top⟶Top
12 ffn 6516 . . . . . 6 (𝑘Gen:Top⟶Top → 𝑘Gen Fn Top)
1311, 12ax-mp 5 . . . . 5 𝑘Gen Fn Top
14 fnfvelrn 6850 . . . . 5 ((𝑘Gen Fn Top ∧ 𝐽 ∈ Top) → (𝑘Gen‘𝐽) ∈ ran 𝑘Gen)
1513, 1, 14sylancr 589 . . . 4 (𝐹 ∈ (𝐽 Cn 𝐾) → (𝑘Gen‘𝐽) ∈ ran 𝑘Gen)
16 cntop2 21851 . . . 4 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
17 kgencn3 22168 . . . 4 (((𝑘Gen‘𝐽) ∈ ran 𝑘Gen ∧ 𝐾 ∈ Top) → ((𝑘Gen‘𝐽) Cn 𝐾) = ((𝑘Gen‘𝐽) Cn (𝑘Gen‘𝐾)))
1815, 16, 17syl2anc 586 . . 3 (𝐹 ∈ (𝐽 Cn 𝐾) → ((𝑘Gen‘𝐽) Cn 𝐾) = ((𝑘Gen‘𝐽) Cn (𝑘Gen‘𝐾)))
1910, 18sseqtrd 4009 . 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))
2119, 20sseldd 3970 1 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹 ∈ ((𝑘Gen‘𝐽) Cn (𝑘Gen‘𝐾)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2114  wss 3938   cuni 4840  ran crn 5558   Fn wfn 6352  wf 6353  cfv 6357  (class class class)co 7158  Topctop 21503  TopOnctopon 21520   Cn ccn 21834  𝑘Genckgen 22143
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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-oadd 8108  df-er 8291  df-map 8410  df-en 8512  df-dom 8513  df-fin 8515  df-fi 8877  df-rest 16698  df-topgen 16719  df-top 21504  df-topon 21521  df-bases 21556  df-cn 21837  df-cmp 21997  df-kgen 22144
This theorem is referenced by: (None)
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