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Theorem qtopid 17396
Description: A quotient map a continuous function into its quotient topology. (Contributed by Mario Carneiro, 23-Mar-2015.)
Assertion
Ref Expression
qtopid  |-  ( ( J  e.  (TopOn `  X )  /\  F  Fn  X )  ->  F  e.  ( J  Cn  ( J qTop  F ) ) )

Proof of Theorem qtopid
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simpr 447 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  F  Fn  X )  ->  F  Fn  X )
2 dffn4 5457 . . . 4  |-  ( F  Fn  X  <->  F : X -onto-> ran  F )
31, 2sylib 188 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  F  Fn  X )  ->  F : X -onto-> ran  F )
4 fof 5451 . . 3  |-  ( F : X -onto-> ran  F  ->  F : X --> ran  F
)
53, 4syl 15 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  F  Fn  X )  ->  F : X --> ran  F )
6 elqtop3 17394 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  F : X -onto-> ran  F )  -> 
( x  e.  ( J qTop  F )  <->  ( x  C_ 
ran  F  /\  ( `' F " x )  e.  J ) ) )
73, 6syldan 456 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  F  Fn  X )  ->  (
x  e.  ( J qTop 
F )  <->  ( x  C_ 
ran  F  /\  ( `' F " x )  e.  J ) ) )
87simplbda 607 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  Fn  X )  /\  x  e.  ( J qTop  F ) )  ->  ( `' F " x )  e.  J )
98ralrimiva 2626 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  F  Fn  X )  ->  A. x  e.  ( J qTop  F ) ( `' F "
x )  e.  J
)
10 qtoptopon 17395 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  F : X -onto-> ran  F )  -> 
( J qTop  F )  e.  (TopOn `  ran  F ) )
113, 10syldan 456 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  F  Fn  X )  ->  ( J qTop  F )  e.  (TopOn `  ran  F ) )
12 iscn 16965 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  ( J qTop  F )  e.  (TopOn `  ran  F ) )  ->  ( F  e.  ( J  Cn  ( J qTop  F ) )  <->  ( F : X --> ran  F  /\  A. x  e.  ( J qTop 
F ) ( `' F " x )  e.  J ) ) )
1311, 12syldan 456 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  F  Fn  X )  ->  ( F  e.  ( J  Cn  ( J qTop  F ) )  <->  ( F : X
--> ran  F  /\  A. x  e.  ( J qTop  F ) ( `' F " x )  e.  J
) ) )
145, 9, 13mpbir2and 888 1  |-  ( ( J  e.  (TopOn `  X )  /\  F  Fn  X )  ->  F  e.  ( J  Cn  ( J qTop  F ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1684   A.wral 2543    C_ wss 3152   `'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  TopOnctopon 16632    Cn ccn 16954
This theorem is referenced by:  qtopcmplem  17398  qtopkgen  17401  qtoprest  17408  kqid  17419  qtopf1  17507  qtophmeo  17508  divstgplem  17803
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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