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Theorem qtopid 19411
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 461 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  F  Fn  X )  ->  F  Fn  X )
2 dffn4 5735 . . . 4  |-  ( F  Fn  X  <->  F : X -onto-> ran  F )
31, 2sylib 196 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  F  Fn  X )  ->  F : X -onto-> ran  F )
4 fof 5729 . . 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 19409 . . . . 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 470 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  F  Fn  X )  ->  (
x  e.  ( J qTop 
F )  <->  ( x  C_ 
ran  F  /\  ( `' F " x )  e.  J ) ) )
87simplbda 624 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  Fn  X )  /\  x  e.  ( J qTop  F ) )  ->  ( `' F " x )  e.  J )
98ralrimiva 2830 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  F  Fn  X )  ->  A. x  e.  ( J qTop  F ) ( `' F "
x )  e.  J
)
10 qtoptopon 19410 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  F : X -onto-> ran  F )  -> 
( J qTop  F )  e.  (TopOn `  ran  F ) )
113, 10syldan 470 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  F  Fn  X )  ->  ( J qTop  F )  e.  (TopOn `  ran  F ) )
12 iscn 18972 . . 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 470 . 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 913 1  |-  ( ( J  e.  (TopOn `  X )  /\  F  Fn  X )  ->  F  e.  ( J  Cn  ( J qTop  F ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1758   A.wral 2799    C_ wss 3437   `'ccnv 4948   ran crn 4950   "cima 4952    Fn wfn 5522   -->wf 5523   -onto->wfo 5525   ` cfv 5527  (class class class)co 6201   qTop cqtop 14561  TopOnctopon 18632    Cn ccn 18961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-map 7327  df-qtop 14565  df-top 18636  df-topon 18639  df-cn 18964
This theorem is referenced by:  qtopcmplem  19413  qtopkgen  19416  qtoprest  19423  kqid  19434  qtopf1  19522  qtophmeo  19523  divstgplem  19824
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