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Theorem qtopss 20508
Description: A surjective continuous function from  J to  K induces a topology  J qTop  F on the base set of  K. This topology is in general finer than  K. Together with qtopid 20498, this implies that  J qTop  F is the finest topology making  F continuous, i.e. the final topology with respect to the family  { F }. (Contributed by Mario Carneiro, 24-Mar-2015.)
Assertion
Ref Expression
qtopss  |-  ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y
)  /\  ran  F  =  Y )  ->  K  C_  ( J qTop  F ) )

Proof of Theorem qtopss
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 toponss 19722 . . . . 5  |-  ( ( K  e.  (TopOn `  Y )  /\  x  e.  K )  ->  x  C_  Y )
213ad2antl2 1160 . . . 4  |-  ( ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y )  /\  ran  F  =  Y )  /\  x  e.  K )  ->  x  C_  Y )
3 cnima 20059 . . . . 5  |-  ( ( F  e.  ( J  Cn  K )  /\  x  e.  K )  ->  ( `' F "
x )  e.  J
)
433ad2antl1 1159 . . . 4  |-  ( ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y )  /\  ran  F  =  Y )  /\  x  e.  K )  ->  ( `' F "
x )  e.  J
)
5 simpl1 1000 . . . . . . 7  |-  ( ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y )  /\  ran  F  =  Y )  /\  x  e.  K )  ->  F  e.  ( J  Cn  K ) )
6 cntop1 20034 . . . . . . 7  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  Top )
75, 6syl 17 . . . . . 6  |-  ( ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y )  /\  ran  F  =  Y )  /\  x  e.  K )  ->  J  e.  Top )
8 eqid 2402 . . . . . . 7  |-  U. J  =  U. J
98toptopon 19726 . . . . . 6  |-  ( J  e.  Top  <->  J  e.  (TopOn `  U. J ) )
107, 9sylib 196 . . . . 5  |-  ( ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y )  /\  ran  F  =  Y )  /\  x  e.  K )  ->  J  e.  (TopOn `  U. J ) )
11 simpl2 1001 . . . . . . . 8  |-  ( ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y )  /\  ran  F  =  Y )  /\  x  e.  K )  ->  K  e.  (TopOn `  Y ) )
12 cnf2 20043 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  U. J )  /\  K  e.  (TopOn `  Y )  /\  F  e.  ( J  Cn  K ) )  ->  F : U. J
--> Y )
1310, 11, 5, 12syl3anc 1230 . . . . . . 7  |-  ( ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y )  /\  ran  F  =  Y )  /\  x  e.  K )  ->  F : U. J --> Y )
14 ffn 5714 . . . . . . 7  |-  ( F : U. J --> Y  ->  F  Fn  U. J )
1513, 14syl 17 . . . . . 6  |-  ( ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y )  /\  ran  F  =  Y )  /\  x  e.  K )  ->  F  Fn  U. J
)
16 simpl3 1002 . . . . . 6  |-  ( ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y )  /\  ran  F  =  Y )  /\  x  e.  K )  ->  ran  F  =  Y )
17 df-fo 5575 . . . . . 6  |-  ( F : U. J -onto-> Y  <->  ( F  Fn  U. J  /\  ran  F  =  Y ) )
1815, 16, 17sylanbrc 662 . . . . 5  |-  ( ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y )  /\  ran  F  =  Y )  /\  x  e.  K )  ->  F : U. J -onto-> Y )
19 elqtop3 20496 . . . . 5  |-  ( ( J  e.  (TopOn `  U. J )  /\  F : U. J -onto-> Y )  ->  ( x  e.  ( J qTop  F )  <-> 
( x  C_  Y  /\  ( `' F "
x )  e.  J
) ) )
2010, 18, 19syl2anc 659 . . . 4  |-  ( ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y )  /\  ran  F  =  Y )  /\  x  e.  K )  ->  ( x  e.  ( J qTop  F )  <->  ( x  C_  Y  /\  ( `' F " x )  e.  J ) ) )
212, 4, 20mpbir2and 923 . . 3  |-  ( ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y )  /\  ran  F  =  Y )  /\  x  e.  K )  ->  x  e.  ( J qTop 
F ) )
2221ex 432 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y
)  /\  ran  F  =  Y )  ->  (
x  e.  K  ->  x  e.  ( J qTop  F ) ) )
2322ssrdv 3448 1  |-  ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y
)  /\  ran  F  =  Y )  ->  K  C_  ( J qTop  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    C_ wss 3414   U.cuni 4191   `'ccnv 4822   ran crn 4824   "cima 4826    Fn wfn 5564   -->wf 5565   -onto->wfo 5567   ` cfv 5569  (class class class)co 6278   qTop cqtop 15117   Topctop 19686  TopOnctopon 19687    Cn ccn 20018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-map 7459  df-qtop 15121  df-top 19691  df-topon 19694  df-cn 20021
This theorem is referenced by:  qtoprest  20510  qtopomap  20511  qtopcmap  20512
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