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Theorem qtopss 19263
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 19253, 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 18509 . . . . 5  |-  ( ( K  e.  (TopOn `  Y )  /\  x  e.  K )  ->  x  C_  Y )
213ad2antl2 1151 . . . 4  |-  ( ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y )  /\  ran  F  =  Y )  /\  x  e.  K )  ->  x  C_  Y )
3 cnima 18844 . . . . 5  |-  ( ( F  e.  ( J  Cn  K )  /\  x  e.  K )  ->  ( `' F "
x )  e.  J
)
433ad2antl1 1150 . . . 4  |-  ( ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y )  /\  ran  F  =  Y )  /\  x  e.  K )  ->  ( `' F "
x )  e.  J
)
5 simpl1 991 . . . . . . 7  |-  ( ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y )  /\  ran  F  =  Y )  /\  x  e.  K )  ->  F  e.  ( J  Cn  K ) )
6 cntop1 18819 . . . . . . 7  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  Top )
75, 6syl 16 . . . . . 6  |-  ( ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y )  /\  ran  F  =  Y )  /\  x  e.  K )  ->  J  e.  Top )
8 eqid 2438 . . . . . . 7  |-  U. J  =  U. J
98toptopon 18513 . . . . . 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 992 . . . . . . . 8  |-  ( ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y )  /\  ran  F  =  Y )  /\  x  e.  K )  ->  K  e.  (TopOn `  Y ) )
12 cnf2 18828 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  U. J )  /\  K  e.  (TopOn `  Y )  /\  F  e.  ( J  Cn  K ) )  ->  F : U. J
--> Y )
1310, 11, 5, 12syl3anc 1218 . . . . . . 7  |-  ( ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y )  /\  ran  F  =  Y )  /\  x  e.  K )  ->  F : U. J --> Y )
14 ffn 5554 . . . . . . 7  |-  ( F : U. J --> Y  ->  F  Fn  U. J )
1513, 14syl 16 . . . . . 6  |-  ( ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y )  /\  ran  F  =  Y )  /\  x  e.  K )  ->  F  Fn  U. J
)
16 simpl3 993 . . . . . 6  |-  ( ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y )  /\  ran  F  =  Y )  /\  x  e.  K )  ->  ran  F  =  Y )
17 df-fo 5419 . . . . . 6  |-  ( F : U. J -onto-> Y  <->  ( F  Fn  U. J  /\  ran  F  =  Y ) )
1815, 16, 17sylanbrc 664 . . . . 5  |-  ( ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y )  /\  ran  F  =  Y )  /\  x  e.  K )  ->  F : U. J -onto-> Y )
19 elqtop3 19251 . . . . 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 661 . . . 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 913 . . 3  |-  ( ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y )  /\  ran  F  =  Y )  /\  x  e.  K )  ->  x  e.  ( J qTop 
F ) )
2221ex 434 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y
)  /\  ran  F  =  Y )  ->  (
x  e.  K  ->  x  e.  ( J qTop  F ) ) )
2322ssrdv 3357 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 369    /\ w3a 965    = wceq 1369    e. wcel 1756    C_ wss 3323   U.cuni 4086   `'ccnv 4834   ran crn 4836   "cima 4838    Fn wfn 5408   -->wf 5409   -onto->wfo 5411   ` cfv 5413  (class class class)co 6086   qTop cqtop 14433   Topctop 18473  TopOnctopon 18474    Cn ccn 18803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-map 7208  df-qtop 14437  df-top 18478  df-topon 18481  df-cn 18806
This theorem is referenced by:  qtoprest  19265  qtopomap  19266  qtopcmap  19267
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