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Theorem qtopf1 17607
Description: If a quotient map is injective, then it is a homeomorphism. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypotheses
Ref Expression
qtopf1.1  |-  ( ph  ->  J  e.  (TopOn `  X ) )
qtopf1.2  |-  ( ph  ->  F : X -1-1-> Y
)
Assertion
Ref Expression
qtopf1  |-  ( ph  ->  F  e.  ( J 
Homeo  ( J qTop  F ) ) )

Proof of Theorem qtopf1
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 qtopf1.1 . . 3  |-  ( ph  ->  J  e.  (TopOn `  X ) )
2 qtopf1.2 . . . 4  |-  ( ph  ->  F : X -1-1-> Y
)
3 f1fn 5518 . . . 4  |-  ( F : X -1-1-> Y  ->  F  Fn  X )
42, 3syl 15 . . 3  |-  ( ph  ->  F  Fn  X )
5 qtopid 17496 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  F  Fn  X )  ->  F  e.  ( J  Cn  ( J qTop  F ) ) )
61, 4, 5syl2anc 642 . 2  |-  ( ph  ->  F  e.  ( J  Cn  ( J qTop  F
) ) )
7 f1f1orn 5563 . . . . 5  |-  ( F : X -1-1-> Y  ->  F : X -1-1-onto-> ran  F )
82, 7syl 15 . . . 4  |-  ( ph  ->  F : X -1-1-onto-> ran  F
)
9 f1ocnv 5565 . . . 4  |-  ( F : X -1-1-onto-> ran  F  ->  `' F : ran  F -1-1-onto-> X )
10 f1of 5552 . . . 4  |-  ( `' F : ran  F -1-1-onto-> X  ->  `' F : ran  F --> X )
118, 9, 103syl 18 . . 3  |-  ( ph  ->  `' F : ran  F --> X )
12 imacnvcnv 5216 . . . . 5  |-  ( `' `' F " x )  =  ( F "
x )
13 imassrn 5104 . . . . . . 7  |-  ( F
" x )  C_  ran  F
1413a1i 10 . . . . . 6  |-  ( (
ph  /\  x  e.  J )  ->  ( F " x )  C_  ran  F )
152adantr 451 . . . . . . . 8  |-  ( (
ph  /\  x  e.  J )  ->  F : X -1-1-> Y )
16 toponss 16767 . . . . . . . . 9  |-  ( ( J  e.  (TopOn `  X )  /\  x  e.  J )  ->  x  C_  X )
171, 16sylan 457 . . . . . . . 8  |-  ( (
ph  /\  x  e.  J )  ->  x  C_  X )
18 f1imacnv 5569 . . . . . . . 8  |-  ( ( F : X -1-1-> Y  /\  x  C_  X )  ->  ( `' F " ( F " x
) )  =  x )
1915, 17, 18syl2anc 642 . . . . . . 7  |-  ( (
ph  /\  x  e.  J )  ->  ( `' F " ( F
" x ) )  =  x )
20 simpr 447 . . . . . . 7  |-  ( (
ph  /\  x  e.  J )  ->  x  e.  J )
2119, 20eqeltrd 2432 . . . . . 6  |-  ( (
ph  /\  x  e.  J )  ->  ( `' F " ( F
" x ) )  e.  J )
22 dffn4 5537 . . . . . . . . 9  |-  ( F  Fn  X  <->  F : X -onto-> ran  F )
234, 22sylib 188 . . . . . . . 8  |-  ( ph  ->  F : X -onto-> ran  F )
24 elqtop3 17494 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  X )  /\  F : X -onto-> ran  F )  -> 
( ( F "
x )  e.  ( J qTop  F )  <->  ( ( F " x )  C_  ran  F  /\  ( `' F " ( F
" x ) )  e.  J ) ) )
251, 23, 24syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( ( F "
x )  e.  ( J qTop  F )  <->  ( ( F " x )  C_  ran  F  /\  ( `' F " ( F
" x ) )  e.  J ) ) )
2625adantr 451 . . . . . 6  |-  ( (
ph  /\  x  e.  J )  ->  (
( F " x
)  e.  ( J qTop 
F )  <->  ( ( F " x )  C_  ran  F  /\  ( `' F " ( F
" x ) )  e.  J ) ) )
2714, 21, 26mpbir2and 888 . . . . 5  |-  ( (
ph  /\  x  e.  J )  ->  ( F " x )  e.  ( J qTop  F ) )
2812, 27syl5eqel 2442 . . . 4  |-  ( (
ph  /\  x  e.  J )  ->  ( `' `' F " x )  e.  ( J qTop  F
) )
2928ralrimiva 2702 . . 3  |-  ( ph  ->  A. x  e.  J  ( `' `' F " x )  e.  ( J qTop  F
) )
30 qtoptopon 17495 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  F : X -onto-> ran  F )  -> 
( J qTop  F )  e.  (TopOn `  ran  F ) )
311, 23, 30syl2anc 642 . . . 4  |-  ( ph  ->  ( J qTop  F )  e.  (TopOn `  ran  F ) )
32 iscn 17065 . . . 4  |-  ( ( ( J qTop  F )  e.  (TopOn `  ran  F )  /\  J  e.  (TopOn `  X )
)  ->  ( `' F  e.  ( ( J qTop  F )  Cn  J
)  <->  ( `' F : ran  F --> X  /\  A. x  e.  J  ( `' `' F " x )  e.  ( J qTop  F
) ) ) )
3331, 1, 32syl2anc 642 . . 3  |-  ( ph  ->  ( `' F  e.  ( ( J qTop  F
)  Cn  J )  <-> 
( `' F : ran  F --> X  /\  A. x  e.  J  ( `' `' F " x )  e.  ( J qTop  F
) ) ) )
3411, 29, 33mpbir2and 888 . 2  |-  ( ph  ->  `' F  e.  (
( J qTop  F )  Cn  J ) )
35 ishmeo 17550 . 2  |-  ( F  e.  ( J  Homeo  ( J qTop  F ) )  <-> 
( F  e.  ( J  Cn  ( J qTop 
F ) )  /\  `' F  e.  (
( J qTop  F )  Cn  J ) ) )
366, 34, 35sylanbrc 645 1  |-  ( ph  ->  F  e.  ( J 
Homeo  ( J qTop  F ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1642    e. wcel 1710   A.wral 2619    C_ wss 3228   `'ccnv 4767   ran crn 4769   "cima 4771    Fn wfn 5329   -->wf 5330   -1-1->wf1 5331   -onto->wfo 5332   -1-1-onto->wf1o 5333   ` cfv 5334  (class class class)co 5942   qTop cqtop 13499  TopOnctopon 16732    Cn ccn 17054    Homeo chmeo 17544
This theorem is referenced by:  t0kq  17609
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-map 6859  df-qtop 13503  df-top 16736  df-topon 16739  df-cn 17057  df-hmeo 17546
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