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Theorem kgen2cn 20651
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  e.  ( J  Cn  K )  ->  F  e.  ( (𝑘Gen `  J )  Cn  (𝑘Gen `  K ) ) )

Proof of Theorem kgen2cn
StepHypRef Expression
1 cntop1 20333 . . . . . 6  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  Top )
2 eqid 2471 . . . . . . 7  |-  U. J  =  U. J
32toptopon 20025 . . . . . 6  |-  ( J  e.  Top  <->  J  e.  (TopOn `  U. J ) )
41, 3sylib 201 . . . . 5  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  (TopOn `  U. J ) )
5 kgentopon 20630 . . . . 5  |-  ( J  e.  (TopOn `  U. J )  ->  (𝑘Gen `  J )  e.  (TopOn `  U. J ) )
64, 5syl 17 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  (𝑘Gen `  J )  e.  (TopOn `  U. J ) )
7 kgenss 20635 . . . . 5  |-  ( J  e.  Top  ->  J  C_  (𝑘Gen `  J ) )
81, 7syl 17 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  J  C_  (𝑘Gen `  J ) )
92cnss1 20369 . . . 4  |-  ( ( (𝑘Gen `  J )  e.  (TopOn `  U. J )  /\  J  C_  (𝑘Gen `  J ) )  -> 
( J  Cn  K
)  C_  ( (𝑘Gen `  J )  Cn  K
) )
106, 8, 9syl2anc 673 . . 3  |-  ( F  e.  ( J  Cn  K )  ->  ( J  Cn  K )  C_  ( (𝑘Gen `  J )  Cn  K ) )
11 kgenf 20633 . . . . . 6  |- 𝑘Gen : Top --> Top
12 ffn 5739 . . . . . 6  |-  (𝑘Gen : Top --> Top 
-> 𝑘Gen 
Fn  Top )
1311, 12ax-mp 5 . . . . 5  |- 𝑘Gen  Fn  Top
14 fnfvelrn 6034 . . . . 5  |-  ( (𝑘Gen  Fn  Top  /\  J  e. 
Top )  ->  (𝑘Gen `  J )  e.  ran 𝑘Gen )
1513, 1, 14sylancr 676 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  (𝑘Gen `  J )  e.  ran 𝑘Gen )
16 cntop2 20334 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
17 kgencn3 20650 . . . 4  |-  ( ( (𝑘Gen `  J )  e. 
ran 𝑘Gen 
/\  K  e.  Top )  ->  ( (𝑘Gen `  J
)  Cn  K )  =  ( (𝑘Gen `  J
)  Cn  (𝑘Gen `  K
) ) )
1815, 16, 17syl2anc 673 . . 3  |-  ( F  e.  ( J  Cn  K )  ->  (
(𝑘Gen `  J )  Cn  K )  =  ( (𝑘Gen `  J )  Cn  (𝑘Gen `  K ) ) )
1910, 18sseqtrd 3454 . 2  |-  ( F  e.  ( J  Cn  K )  ->  ( J  Cn  K )  C_  ( (𝑘Gen `  J )  Cn  (𝑘Gen `  K ) ) )
20 id 22 . 2  |-  ( F  e.  ( J  Cn  K )  ->  F  e.  ( J  Cn  K
) )
2119, 20sseldd 3419 1  |-  ( F  e.  ( J  Cn  K )  ->  F  e.  ( (𝑘Gen `  J )  Cn  (𝑘Gen `  K ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1452    e. wcel 1904    C_ wss 3390   U.cuni 4190   ran crn 4840    Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308   Topctop 19994  TopOnctopon 19995    Cn ccn 20317  𝑘Genckgen 20625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-fin 7591  df-fi 7943  df-rest 15399  df-topgen 15420  df-top 19998  df-bases 19999  df-topon 20000  df-cn 20320  df-cmp 20479  df-kgen 20626
This theorem is referenced by: (None)
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