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Theorem qtopid 17729
Description: A quotient map is 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 448 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  F  Fn  X )  ->  F  Fn  X )
2 dffn4 5651 . . . 4  |-  ( F  Fn  X  <->  F : X -onto-> ran  F )
31, 2sylib 189 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  F  Fn  X )  ->  F : X -onto-> ran  F )
4 fof 5645 . . 3  |-  ( F : X -onto-> ran  F  ->  F : X --> ran  F
)
53, 4syl 16 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  F  Fn  X )  ->  F : X --> ran  F )
6 elqtop3 17727 . . . . 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 457 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  F  Fn  X )  ->  (
x  e.  ( J qTop 
F )  <->  ( x  C_ 
ran  F  /\  ( `' F " x )  e.  J ) ) )
87simplbda 608 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  Fn  X )  /\  x  e.  ( J qTop  F ) )  ->  ( `' F " x )  e.  J )
98ralrimiva 2781 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  F  Fn  X )  ->  A. x  e.  ( J qTop  F ) ( `' F "
x )  e.  J
)
10 qtoptopon 17728 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  F : X -onto-> ran  F )  -> 
( J qTop  F )  e.  (TopOn `  ran  F ) )
113, 10syldan 457 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  F  Fn  X )  ->  ( J qTop  F )  e.  (TopOn `  ran  F ) )
12 iscn 17291 . . 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 457 . 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 889 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 177    /\ wa 359    e. wcel 1725   A.wral 2697    C_ wss 3312   `'ccnv 4869   ran crn 4871   "cima 4873    Fn wfn 5441   -->wf 5442   -onto->wfo 5444   ` cfv 5446  (class class class)co 6073   qTop cqtop 13721  TopOnctopon 16951    Cn ccn 17280
This theorem is referenced by:  qtopcmplem  17731  qtopkgen  17734  qtoprest  17741  kqid  17752  qtopf1  17840  qtophmeo  17841  divstgplem  18142
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-map 7012  df-qtop 13725  df-top 16955  df-topon 16958  df-cn 17283
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