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Theorem qtopkgen 20079
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 19911 . . 3  |-  ( J  e.  ran 𝑘Gen  ->  J  e.  Top )
2 qtopcmp.1 . . . 4  |-  X  = 
U. J
32qtoptop 20069 . . 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 4281 . . . . . . . 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 2467 . . . . . . . . 9  |-  U. ( J qTop  F )  =  U. ( J qTop  F )
98kgenuni 19908 . . . . . . . 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 3546 . . . . . 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 5807 . . . . . . . 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 20071 . . . . . . 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 3546 . . . . 5  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  x  C_ 
ran  F )
202toptopon 19303 . . . . . . . . 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 20074 . . . . . . . 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 19927 . . . . . . . 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 2557 . . . . . 6  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  F  e.  ( J  Cn  (𝑘Gen `  ( J qTop  F )
) ) )
27 cnima 19634 . . . . . 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 20070 . . . . . 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 920 . . . 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 3515 . 2  |-  ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  ->  (𝑘Gen `  ( J qTop  F )
)  C_  ( J qTop  F ) )
34 iskgen2 19917 . 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 1379    e. wcel 1767    C_ wss 3481   U.cuni 4251   `'ccnv 5004   ran crn 5006   "cima 5008    Fn wfn 5589   -onto->wfo 5592   ` cfv 5594  (class class class)co 6295   qTop cqtop 14775   Topctop 19263  TopOnctopon 19264    Cn ccn 19593  𝑘Genckgen 19902
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-fin 7532  df-fi 7883  df-rest 14695  df-topgen 14716  df-qtop 14779  df-top 19268  df-bases 19270  df-topon 19271  df-cn 19596  df-cmp 19755  df-kgen 19903
This theorem is referenced by: (None)
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