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Theorem qtopkgen 17401
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 17237 . . 3  |-  ( J  e.  ran 𝑘Gen  ->  J  e.  Top )
2 qtopcmp.1 . . . 4  |-  X  = 
U. J
32qtoptop 17391 . . 3  |-  ( ( J  e.  Top  /\  F  Fn  X )  ->  ( J qTop  F )  e.  Top )
41, 3sylan 457 . 2  |-  ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  ->  ( J qTop  F )  e.  Top )
5 elssuni 3855 . . . . . . . 8  |-  ( x  e.  (𝑘Gen `  ( J qTop  F
) )  ->  x  C_ 
U. (𝑘Gen `  ( J qTop  F
) ) )
65adantl 452 . . . . . . 7  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  x  C_ 
U. (𝑘Gen `  ( J qTop  F
) ) )
74adantr 451 . . . . . . . 8  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  ( J qTop  F )  e.  Top )
8 eqid 2283 . . . . . . . . 9  |-  U. ( J qTop  F )  =  U. ( J qTop  F )
98kgenuni 17234 . . . . . . . 8  |-  ( ( J qTop  F )  e. 
Top  ->  U. ( J qTop  F
)  =  U. (𝑘Gen `  ( J qTop  F )
) )
107, 9syl 15 . . . . . . 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 3215 . . . . . 6  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  x  C_ 
U. ( J qTop  F
) )
12 simpll 730 . . . . . . . 8  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  J  e.  ran 𝑘Gen )
1312, 1syl 15 . . . . . . 7  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  J  e.  Top )
14 simplr 731 . . . . . . . 8  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  F  Fn  X )
15 dffn4 5457 . . . . . . . 8  |-  ( F  Fn  X  <->  F : X -onto-> ran  F )
1614, 15sylib 188 . . . . . . 7  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  F : X -onto-> ran  F )
172qtopuni 17393 . . . . . . 7  |-  ( ( J  e.  Top  /\  F : X -onto-> ran  F
)  ->  ran  F  = 
U. ( J qTop  F
) )
1813, 16, 17syl2anc 642 . . . . . 6  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  ran  F  =  U. ( J qTop 
F ) )
1911, 18sseqtr4d 3215 . . . . 5  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  x  C_ 
ran  F )
202toptopon 16671 . . . . . . . . 9  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
2113, 20sylib 188 . . . . . . . 8  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  J  e.  (TopOn `  X )
)
22 qtopid 17396 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  X )  /\  F  Fn  X )  ->  F  e.  ( J  Cn  ( J qTop  F ) ) )
2321, 14, 22syl2anc 642 . . . . . . 7  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  F  e.  ( J  Cn  ( J qTop  F ) ) )
24 kgencn3 17253 . . . . . . . 8  |-  ( ( J  e.  ran 𝑘Gen  /\  ( J qTop  F )  e.  Top )  ->  ( J  Cn  ( J qTop  F )
)  =  ( J  Cn  (𝑘Gen `  ( J qTop  F
) ) ) )
2512, 7, 24syl2anc 642 . . . . . . 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 2359 . . . . . 6  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  F  e.  ( J  Cn  (𝑘Gen `  ( J qTop  F )
) ) )
27 cnima 16994 . . . . . 6  |-  ( ( F  e.  ( J  Cn  (𝑘Gen `  ( J qTop  F
) ) )  /\  x  e.  (𝑘Gen `  ( J qTop  F ) ) )  ->  ( `' F " x )  e.  J
)
2826, 27sylancom 648 . . . . 5  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  ( `' F " x )  e.  J )
292elqtop2 17392 . . . . . 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 642 . . . . 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 888 . . . 4  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  x  e.  ( J qTop  F ) )
3231ex 423 . . 3  |-  ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  ->  (
x  e.  (𝑘Gen `  ( J qTop  F ) )  ->  x  e.  ( J qTop  F ) ) )
3332ssrdv 3185 . 2  |-  ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  ->  (𝑘Gen `  ( J qTop  F )
)  C_  ( J qTop  F ) )
34 iskgen2 17243 . 2  |-  ( ( J qTop  F )  e. 
ran 𝑘Gen  <-> 
( ( J qTop  F
)  e.  Top  /\  (𝑘Gen
`  ( J qTop  F
) )  C_  ( J qTop  F ) ) )
354, 33, 34sylanbrc 645 1  |-  ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  ->  ( J qTop  F )  e.  ran 𝑘Gen )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    C_ wss 3152   U.cuni 3827   `'ccnv 4688   ran crn 4690   "cima 4692    Fn wfn 5250   -onto->wfo 5253   ` cfv 5255  (class class class)co 5858   qTop cqtop 13406   Topctop 16631  TopOnctopon 16632    Cn ccn 16954  𝑘Genckgen 17228
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-fin 6867  df-fi 7165  df-rest 13327  df-topgen 13344  df-qtop 13410  df-top 16636  df-bases 16638  df-topon 16639  df-cn 16957  df-cmp 17114  df-kgen 17229
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