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Theorem qtopss 17406
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 17396, 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 16667 . . . . 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 16994 . . . . 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 16970 . . . . . . 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 2283 . . . . . . 7  |-  U. J  =  U. J
98toptopon 16671 . . . . . 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 16979 . . . . . . . 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 5389 . . . . . . 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 5261 . . . . . 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 17394 . . . . 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 3185 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 1623    e. wcel 1684    C_ wss 3152   U.cuni 3827   `'ccnv 4688   ran crn 4690   "cima 4692    Fn wfn 5250   -->wf 5251   -onto->wfo 5253   ` cfv 5255  (class class class)co 5858   qTop cqtop 13406   Topctop 16631  TopOnctopon 16632    Cn ccn 16954
This theorem is referenced by:  qtoprest  17408  qtopomap  17409  qtopcmap  17410
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-qtop 13410  df-top 16636  df-topon 16639  df-cn 16957
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