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Theorem qtopkgen 20377
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 20209 . . 3  |-  ( J  e.  ran 𝑘Gen  ->  J  e.  Top )
2 qtopcmp.1 . . . 4  |-  X  = 
U. J
32qtoptop 20367 . . 3  |-  ( ( J  e.  Top  /\  F  Fn  X )  ->  ( J qTop  F )  e.  Top )
41, 3sylan 469 . 2  |-  ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  ->  ( J qTop  F )  e.  Top )
5 elssuni 4264 . . . . . . . 8  |-  ( x  e.  (𝑘Gen `  ( J qTop  F
) )  ->  x  C_ 
U. (𝑘Gen `  ( J qTop  F
) ) )
65adantl 464 . . . . . . 7  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  x  C_ 
U. (𝑘Gen `  ( J qTop  F
) ) )
74adantr 463 . . . . . . . 8  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  ( J qTop  F )  e.  Top )
8 eqid 2454 . . . . . . . . 9  |-  U. ( J qTop  F )  =  U. ( J qTop  F )
98kgenuni 20206 . . . . . . . 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 3526 . . . . . 6  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  x  C_ 
U. ( J qTop  F
) )
12 simpll 751 . . . . . . . 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 753 . . . . . . . 8  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  F  Fn  X )
15 dffn4 5783 . . . . . . . 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 20369 . . . . . . 7  |-  ( ( J  e.  Top  /\  F : X -onto-> ran  F
)  ->  ran  F  = 
U. ( J qTop  F
) )
1813, 16, 17syl2anc 659 . . . . . 6  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  ran  F  =  U. ( J qTop 
F ) )
1911, 18sseqtr4d 3526 . . . . 5  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  x  C_ 
ran  F )
202toptopon 19601 . . . . . . . . 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 20372 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  X )  /\  F  Fn  X )  ->  F  e.  ( J  Cn  ( J qTop  F ) ) )
2321, 14, 22syl2anc 659 . . . . . . 7  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  F  e.  ( J  Cn  ( J qTop  F ) ) )
24 kgencn3 20225 . . . . . . . 8  |-  ( ( J  e.  ran 𝑘Gen  /\  ( J qTop  F )  e.  Top )  ->  ( J  Cn  ( J qTop  F )
)  =  ( J  Cn  (𝑘Gen `  ( J qTop  F
) ) ) )
2512, 7, 24syl2anc 659 . . . . . . 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 2544 . . . . . 6  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  F  e.  ( J  Cn  (𝑘Gen `  ( J qTop  F )
) ) )
27 cnima 19933 . . . . . 6  |-  ( ( F  e.  ( J  Cn  (𝑘Gen `  ( J qTop  F
) ) )  /\  x  e.  (𝑘Gen `  ( J qTop  F ) ) )  ->  ( `' F " x )  e.  J
)
2826, 27sylancom 665 . . . . 5  |-  ( ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  /\  x  e.  (𝑘Gen `  ( J qTop  F )
) )  ->  ( `' F " x )  e.  J )
292elqtop2 20368 . . . . . 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 659 . . . . 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 432 . . 3  |-  ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  ->  (
x  e.  (𝑘Gen `  ( J qTop  F ) )  ->  x  e.  ( J qTop  F ) ) )
3332ssrdv 3495 . 2  |-  ( ( J  e.  ran 𝑘Gen  /\  F  Fn  X )  ->  (𝑘Gen `  ( J qTop  F )
)  C_  ( J qTop  F ) )
34 iskgen2 20215 . 2  |-  ( ( J qTop  F )  e. 
ran 𝑘Gen  <-> 
( ( J qTop  F
)  e.  Top  /\  (𝑘Gen
`  ( J qTop  F
) )  C_  ( J qTop  F ) ) )
354, 33, 34sylanbrc 662 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 367    = wceq 1398    e. wcel 1823    C_ wss 3461   U.cuni 4235   `'ccnv 4987   ran crn 4989   "cima 4991    Fn wfn 5565   -onto->wfo 5568   ` cfv 5570  (class class class)co 6270   qTop cqtop 14992   Topctop 19561  TopOnctopon 19562    Cn ccn 19892  𝑘Genckgen 20200
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-fin 7513  df-fi 7863  df-rest 14912  df-topgen 14933  df-qtop 14996  df-top 19566  df-bases 19568  df-topon 19569  df-cn 19895  df-cmp 20054  df-kgen 20201
This theorem is referenced by: (None)
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