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Theorem qtopss 17749
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 17739, 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 16996 . . . . 5  |-  ( ( K  e.  (TopOn `  Y )  /\  x  e.  K )  ->  x  C_  Y )
213ad2antl2 1121 . . . 4  |-  ( ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y )  /\  ran  F  =  Y )  /\  x  e.  K )  ->  x  C_  Y )
3 cnima 17331 . . . . 5  |-  ( ( F  e.  ( J  Cn  K )  /\  x  e.  K )  ->  ( `' F "
x )  e.  J
)
433ad2antl1 1120 . . . 4  |-  ( ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y )  /\  ran  F  =  Y )  /\  x  e.  K )  ->  ( `' F "
x )  e.  J
)
5 simpl1 961 . . . . . . 7  |-  ( ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y )  /\  ran  F  =  Y )  /\  x  e.  K )  ->  F  e.  ( J  Cn  K ) )
6 cntop1 17306 . . . . . . 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 17000 . . . . . 6  |-  ( J  e.  Top  <->  J  e.  (TopOn `  U. J ) )
107, 9sylib 190 . . . . 5  |-  ( ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y )  /\  ran  F  =  Y )  /\  x  e.  K )  ->  J  e.  (TopOn `  U. J ) )
11 simpl2 962 . . . . . . . 8  |-  ( ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y )  /\  ran  F  =  Y )  /\  x  e.  K )  ->  K  e.  (TopOn `  Y ) )
12 cnf2 17315 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  U. J )  /\  K  e.  (TopOn `  Y )  /\  F  e.  ( J  Cn  K ) )  ->  F : U. J
--> Y )
1310, 11, 5, 12syl3anc 1185 . . . . . . 7  |-  ( ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y )  /\  ran  F  =  Y )  /\  x  e.  K )  ->  F : U. J --> Y )
14 ffn 5593 . . . . . . 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 963 . . . . . 6  |-  ( ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y )  /\  ran  F  =  Y )  /\  x  e.  K )  ->  ran  F  =  Y )
17 df-fo 5462 . . . . . 6  |-  ( F : U. J -onto-> Y  <->  ( F  Fn  U. J  /\  ran  F  =  Y ) )
1815, 16, 17sylanbrc 647 . . . . 5  |-  ( ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y )  /\  ran  F  =  Y )  /\  x  e.  K )  ->  F : U. J -onto-> Y )
19 elqtop3 17737 . . . . 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 644 . . . 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 890 . . 3  |-  ( ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y )  /\  ran  F  =  Y )  /\  x  e.  K )  ->  x  e.  ( J qTop 
F ) )
2221ex 425 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y
)  /\  ran  F  =  Y )  ->  (
x  e.  K  ->  x  e.  ( J qTop  F ) ) )
2322ssrdv 3356 1  |-  ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y
)  /\  ran  F  =  Y )  ->  K  C_  ( J qTop  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    C_ wss 3322   U.cuni 4017   `'ccnv 4879   ran crn 4881   "cima 4883    Fn wfn 5451   -->wf 5452   -onto->wfo 5454   ` cfv 5456  (class class class)co 6083   qTop cqtop 13731   Topctop 16960  TopOnctopon 16961    Cn ccn 17290
This theorem is referenced by:  qtoprest  17751  qtopomap  17752  qtopcmap  17753
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 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-map 7022  df-qtop 13735  df-top 16965  df-topon 16968  df-cn 17293
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