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Theorem qtopkgen 19283
Description: A quotient of a compactly generated space is compactly generated. (Contributed by Mario Carneiro, 9-Apr-2015.)
Hypothesis
Ref Expression
qtopcmp.1  |-  X  = 
U. J
Assertion
Ref Expression
qtopkgen  |-  ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  ->  ( J qTop  F )  e.  ran 𝑘Gen )

Proof of Theorem qtopkgen
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 kgentop 19115 . . 3  |-  ( J  e.  ran 𝑘Gen  ->  J  e.  Top )
2 qtopcmp.1 . . . 4  |-  X  = 
U. J
32qtoptop 19273 . . 3  |-  ( ( J  e.  Top  /\  F  Fn  X )  ->  ( J qTop  F )  e.  Top )
41, 3sylan 471 . 2  |-  ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  ->  ( J qTop  F )  e.  Top )
5 elssuni 4121 . . . . . . . 8  |-  ( x  e.  (𝑘Gen `  ( J qTop  F
) )  ->  x  C_ 
U. (𝑘Gen `  ( J qTop  F
) ) )
65adantl 466 . . . . . . 7  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  x  C_ 
U. (𝑘Gen `  ( J qTop  F
) ) )
74adantr 465 . . . . . . . 8  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  ( J qTop  F )  e.  Top )
8 eqid 2443 . . . . . . . . 9  |-  U. ( J qTop  F )  =  U. ( J qTop  F )
98kgenuni 19112 . . . . . . . 8  |-  ( ( J qTop  F )  e. 
Top  ->  U. ( J qTop  F
)  =  U. (𝑘Gen `  ( J qTop  F )
) )
107, 9syl 16 . . . . . . 7  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  U. ( J qTop  F )  =  U. (𝑘Gen
`  ( J qTop  F
) ) )
116, 10sseqtr4d 3393 . . . . . 6  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  x  C_ 
U. ( J qTop  F
) )
12 simpll 753 . . . . . . . 8  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  J  e.  ran 𝑘Gen )
1312, 1syl 16 . . . . . . 7  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  J  e.  Top )
14 simplr 754 . . . . . . . 8  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  F  Fn  X )
15 dffn4 5626 . . . . . . . 8  |-  ( F  Fn  X  <->  F : X -onto-> ran  F )
1614, 15sylib 196 . . . . . . 7  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  F : X -onto-> ran  F )
172qtopuni 19275 . . . . . . 7  |-  ( ( J  e.  Top  /\  F : X -onto-> ran  F
)  ->  ran  F  = 
U. ( J qTop  F
) )
1813, 16, 17syl2anc 661 . . . . . 6  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  ran  F  =  U. ( J qTop 
F ) )
1911, 18sseqtr4d 3393 . . . . 5  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  x  C_ 
ran  F )
202toptopon 18538 . . . . . . . . 9  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
2113, 20sylib 196 . . . . . . . 8  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  J  e.  (TopOn `  X )
)
22 qtopid 19278 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  X )  /\  F  Fn  X )  ->  F  e.  ( J  Cn  ( J qTop  F ) ) )
2321, 14, 22syl2anc 661 . . . . . . 7  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  F  e.  ( J  Cn  ( J qTop  F ) ) )
24 kgencn3 19131 . . . . . . . 8  |-  ( ( J  e.  ran 𝑘Gen  /\  ( J qTop  F )  e.  Top )  ->  ( J  Cn  ( J qTop  F )
)  =  ( J  Cn  (𝑘Gen `  ( J qTop  F
) ) ) )
2512, 7, 24syl2anc 661 . . . . . . 7  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  ( J  Cn  ( J qTop  F
) )  =  ( J  Cn  (𝑘Gen `  ( J qTop  F ) ) ) )
2623, 25eleqtrd 2519 . . . . . 6  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  F  e.  ( J  Cn  (𝑘Gen `  ( J qTop  F )
) ) )
27 cnima 18869 . . . . . 6  |-  ( ( F  e.  ( J  Cn  (𝑘Gen `  ( J qTop  F
) ) )  /\  x  e.  (𝑘Gen `  ( J qTop  F ) ) )  ->  ( `' F " x )  e.  J
)
2826, 27sylancom 667 . . . . 5  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  ( `' F " x )  e.  J )
292elqtop2 19274 . . . . . 6  |-  ( ( J  e.  ran 𝑘Gen  /\  F : X -onto-> ran  F )  -> 
( x  e.  ( J qTop  F )  <->  ( x  C_ 
ran  F  /\  ( `' F " x )  e.  J ) ) )
3012, 16, 29syl2anc 661 . . . . 5  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  (
x  e.  ( J qTop 
F )  <->  ( x  C_ 
ran  F  /\  ( `' F " x )  e.  J ) ) )
3119, 28, 30mpbir2and 913 . . . 4  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  x  e.  ( J qTop  F ) )
3231ex 434 . . 3  |-  ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  ->  (
x  e.  (𝑘Gen `  ( J qTop  F ) )  ->  x  e.  ( J qTop  F ) ) )
3332ssrdv 3362 . 2  |-  ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  ->  (𝑘Gen `  ( J qTop  F )
)  C_  ( J qTop  F ) )
34 iskgen2 19121 . 2  |-  ( ( J qTop  F )  e. 
ran 𝑘Gen  <-> 
( ( J qTop  F
)  e.  Top  /\  (𝑘Gen
`  ( J qTop  F
) )  C_  ( J qTop  F ) ) )
354, 33, 34sylanbrc 664 1  |-  ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  ->  ( J qTop  F )  e.  ran 𝑘Gen )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    C_ wss 3328   U.cuni 4091   `'ccnv 4839   ran crn 4841   "cima 4843    Fn wfn 5413   -onto->wfo 5416   ` cfv 5418  (class class class)co 6091   qTop cqtop 14441   Topctop 18498  TopOnctopon 18499    Cn ccn 18828  𝑘Genckgen 19106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-map 7216  df-en 7311  df-dom 7312  df-fin 7314  df-fi 7661  df-rest 14361  df-topgen 14382  df-qtop 14445  df-top 18503  df-bases 18505  df-topon 18506  df-cn 18831  df-cmp 18990  df-kgen 19107
This theorem is referenced by: (None)
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