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Theorem qtopss 17422
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 17412, 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 16683 . . . . 5  |-  ( ( K  e.  (TopOn `  Y )  /\  x  e.  K )  ->  x  C_  Y )
213ad2antl2 1118 . . . 4  |-  ( ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y )  /\  ran  F  =  Y )  /\  x  e.  K )  ->  x  C_  Y )
3 cnima 17010 . . . . 5  |-  ( ( F  e.  ( J  Cn  K )  /\  x  e.  K )  ->  ( `' F "
x )  e.  J
)
433ad2antl1 1117 . . . 4  |-  ( ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y )  /\  ran  F  =  Y )  /\  x  e.  K )  ->  ( `' F "
x )  e.  J
)
5 simpl1 958 . . . . . . 7  |-  ( ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y )  /\  ran  F  =  Y )  /\  x  e.  K )  ->  F  e.  ( J  Cn  K ) )
6 cntop1 16986 . . . . . . 7  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  Top )
75, 6syl 15 . . . . . 6  |-  ( ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y )  /\  ran  F  =  Y )  /\  x  e.  K )  ->  J  e.  Top )
8 eqid 2296 . . . . . . 7  |-  U. J  =  U. J
98toptopon 16687 . . . . . 6  |-  ( J  e.  Top  <->  J  e.  (TopOn `  U. J ) )
107, 9sylib 188 . . . . 5  |-  ( ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y )  /\  ran  F  =  Y )  /\  x  e.  K )  ->  J  e.  (TopOn `  U. J ) )
11 simpl2 959 . . . . . . . 8  |-  ( ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y )  /\  ran  F  =  Y )  /\  x  e.  K )  ->  K  e.  (TopOn `  Y ) )
12 cnf2 16995 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  U. J )  /\  K  e.  (TopOn `  Y )  /\  F  e.  ( J  Cn  K ) )  ->  F : U. J
--> Y )
1310, 11, 5, 12syl3anc 1182 . . . . . . 7  |-  ( ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y )  /\  ran  F  =  Y )  /\  x  e.  K )  ->  F : U. J --> Y )
14 ffn 5405 . . . . . . 7  |-  ( F : U. J --> Y  ->  F  Fn  U. J )
1513, 14syl 15 . . . . . 6  |-  ( ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y )  /\  ran  F  =  Y )  /\  x  e.  K )  ->  F  Fn  U. J
)
16 simpl3 960 . . . . . 6  |-  ( ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y )  /\  ran  F  =  Y )  /\  x  e.  K )  ->  ran  F  =  Y )
17 df-fo 5277 . . . . . 6  |-  ( F : U. J -onto-> Y  <->  ( F  Fn  U. J  /\  ran  F  =  Y ) )
1815, 16, 17sylanbrc 645 . . . . 5  |-  ( ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y )  /\  ran  F  =  Y )  /\  x  e.  K )  ->  F : U. J -onto-> Y )
19 elqtop3 17410 . . . . 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 642 . . . 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 888 . . 3  |-  ( ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y )  /\  ran  F  =  Y )  /\  x  e.  K )  ->  x  e.  ( J qTop 
F ) )
2221ex 423 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  K  e.  (TopOn `  Y
)  /\  ran  F  =  Y )  ->  (
x  e.  K  ->  x  e.  ( J qTop  F ) ) )
2322ssrdv 3198 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    C_ wss 3165   U.cuni 3843   `'ccnv 4704   ran crn 4706   "cima 4708    Fn wfn 5266   -->wf 5267   -onto->wfo 5269   ` cfv 5271  (class class class)co 5874   qTop cqtop 13422   Topctop 16647  TopOnctopon 16648    Cn ccn 16970
This theorem is referenced by:  qtoprest  17424  qtopomap  17425  qtopcmap  17426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-qtop 13426  df-top 16652  df-topon 16655  df-cn 16973
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