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Theorem qtopid 17412
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 447 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  F  Fn  X )  ->  F  Fn  X )
2 dffn4 5473 . . . 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 5467 . . 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 17410 . . . . 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 2639 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  F  Fn  X )  ->  A. x  e.  ( J qTop  F ) ( `' F "
x )  e.  J
)
10 qtoptopon 17411 . . . 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 16981 . . 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 1696   A.wral 2556    C_ wss 3165   `'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  TopOnctopon 16648    Cn ccn 16970
This theorem is referenced by:  qtopcmplem  17414  qtopkgen  17417  qtoprest  17424  kqid  17435  qtopf1  17523  qtophmeo  17524  divstgplem  17819
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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