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Theorem qtopkgen 17743
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 17575 . . 3  |-  ( J  e.  ran 𝑘Gen  ->  J  e.  Top )
2 qtopcmp.1 . . . 4  |-  X  = 
U. J
32qtoptop 17733 . . 3  |-  ( ( J  e.  Top  /\  F  Fn  X )  ->  ( J qTop  F )  e.  Top )
41, 3sylan 459 . 2  |-  ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  ->  ( J qTop  F )  e.  Top )
5 elssuni 4044 . . . . . . . 8  |-  ( x  e.  (𝑘Gen `  ( J qTop  F
) )  ->  x  C_ 
U. (𝑘Gen `  ( J qTop  F
) ) )
65adantl 454 . . . . . . 7  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  x  C_ 
U. (𝑘Gen `  ( J qTop  F
) ) )
74adantr 453 . . . . . . . 8  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  ( J qTop  F )  e.  Top )
8 eqid 2437 . . . . . . . . 9  |-  U. ( J qTop  F )  =  U. ( J qTop  F )
98kgenuni 17572 . . . . . . . 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 3386 . . . . . 6  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  x  C_ 
U. ( J qTop  F
) )
12 simpll 732 . . . . . . . 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 733 . . . . . . . 8  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  F  Fn  X )
15 dffn4 5660 . . . . . . . 8  |-  ( F  Fn  X  <->  F : X -onto-> ran  F )
1614, 15sylib 190 . . . . . . 7  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  F : X -onto-> ran  F )
172qtopuni 17735 . . . . . . 7  |-  ( ( J  e.  Top  /\  F : X -onto-> ran  F
)  ->  ran  F  = 
U. ( J qTop  F
) )
1813, 16, 17syl2anc 644 . . . . . 6  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  ran  F  =  U. ( J qTop 
F ) )
1911, 18sseqtr4d 3386 . . . . 5  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  x  C_ 
ran  F )
202toptopon 16999 . . . . . . . . 9  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
2113, 20sylib 190 . . . . . . . 8  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  J  e.  (TopOn `  X )
)
22 qtopid 17738 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  X )  /\  F  Fn  X )  ->  F  e.  ( J  Cn  ( J qTop  F ) ) )
2321, 14, 22syl2anc 644 . . . . . . 7  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  F  e.  ( J  Cn  ( J qTop  F ) ) )
24 kgencn3 17591 . . . . . . . 8  |-  ( ( J  e.  ran 𝑘Gen  /\  ( J qTop  F )  e.  Top )  ->  ( J  Cn  ( J qTop  F )
)  =  ( J  Cn  (𝑘Gen `  ( J qTop  F
) ) ) )
2512, 7, 24syl2anc 644 . . . . . . 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 2513 . . . . . 6  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  F  e.  ( J  Cn  (𝑘Gen `  ( J qTop  F )
) ) )
27 cnima 17330 . . . . . 6  |-  ( ( F  e.  ( J  Cn  (𝑘Gen `  ( J qTop  F
) ) )  /\  x  e.  (𝑘Gen `  ( J qTop  F ) ) )  ->  ( `' F " x )  e.  J
)
2826, 27sylancom 650 . . . . 5  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  ( `' F " x )  e.  J )
292elqtop2 17734 . . . . . 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 644 . . . . 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 890 . . . 4  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  x  e.  ( J qTop  F ) )
3231ex 425 . . 3  |-  ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  ->  (
x  e.  (𝑘Gen `  ( J qTop  F ) )  ->  x  e.  ( J qTop  F ) ) )
3332ssrdv 3355 . 2  |-  ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  ->  (𝑘Gen `  ( J qTop  F )
)  C_  ( J qTop  F ) )
34 iskgen2 17581 . 2  |-  ( ( J qTop  F )  e. 
ran 𝑘Gen  <-> 
( ( J qTop  F
)  e.  Top  /\  (𝑘Gen
`  ( J qTop  F
) )  C_  ( J qTop  F ) ) )
354, 33, 34sylanbrc 647 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 178    /\ wa 360    = wceq 1653    e. wcel 1726    C_ wss 3321   U.cuni 4016   `'ccnv 4878   ran crn 4880   "cima 4882    Fn wfn 5450   -onto->wfo 5453   ` cfv 5455  (class class class)co 6082   qTop cqtop 13730   Topctop 16959  TopOnctopon 16960    Cn ccn 17289  𝑘Genckgen 17566
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-recs 6634  df-rdg 6669  df-1o 6725  df-oadd 6729  df-er 6906  df-map 7021  df-en 7111  df-dom 7112  df-fin 7114  df-fi 7417  df-rest 13651  df-topgen 13668  df-qtop 13734  df-top 16964  df-bases 16966  df-topon 16967  df-cn 17292  df-cmp 17451  df-kgen 17567
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