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Theorem qtopid 20033
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 5801 . . . 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 5795 . . 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 20031 . . . . 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 2878 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  F  Fn  X )  ->  A. x  e.  ( J qTop  F ) ( `' F "
x )  e.  J
)
10 qtoptopon 20032 . . . 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 19542 . . 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 920 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 1767   A.wral 2814    C_ wss 3476   `'ccnv 4998   ran crn 5000   "cima 5002    Fn wfn 5583   -->wf 5584   -onto->wfo 5586   ` cfv 5588  (class class class)co 6285   qTop cqtop 14761  TopOnctopon 19202    Cn ccn 19531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-map 7423  df-qtop 14765  df-top 19206  df-topon 19209  df-cn 19534
This theorem is referenced by:  qtopcmplem  20035  qtopkgen  20038  qtoprest  20045  kqid  20056  qtopf1  20144  qtophmeo  20145  qustgplem  20446  circcn  27601
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