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Theorem qtopf1 17509
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 5440 . . . 4  |-  ( F : X -1-1-> Y  ->  F  Fn  X )
42, 3syl 15 . . 3  |-  ( ph  ->  F  Fn  X )
5 qtopid 17398 . . 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 5485 . . . . 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 5487 . . . 4  |-  ( F : X -1-1-onto-> ran  F  ->  `' F : ran  F -1-1-onto-> X )
10 f1of 5474 . . . 4  |-  ( `' F : ran  F -1-1-onto-> X  ->  `' F : ran  F --> X )
118, 9, 103syl 18 . . 3  |-  ( ph  ->  `' F : ran  F --> X )
12 imacnvcnv 5139 . . . . 5  |-  ( `' `' F " x )  =  ( F "
x )
13 imassrn 5027 . . . . . . 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 16669 . . . . . . . . 9  |-  ( ( J  e.  (TopOn `  X )  /\  x  e.  J )  ->  x  C_  X )
171, 16sylan 457 . . . . . . . 8  |-  ( (
ph  /\  x  e.  J )  ->  x  C_  X )
18 f1imacnv 5491 . . . . . . . 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 2359 . . . . . 6  |-  ( (
ph  /\  x  e.  J )  ->  ( `' F " ( F
" x ) )  e.  J )
22 dffn4 5459 . . . . . . . . 9  |-  ( F  Fn  X  <->  F : X -onto-> ran  F )
234, 22sylib 188 . . . . . . . 8  |-  ( ph  ->  F : X -onto-> ran  F )
24 elqtop3 17396 . . . . . . . 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 2369 . . . 4  |-  ( (
ph  /\  x  e.  J )  ->  ( `' `' F " x )  e.  ( J qTop  F
) )
2928ralrimiva 2628 . . 3  |-  ( ph  ->  A. x  e.  J  ( `' `' F " x )  e.  ( J qTop  F
) )
30 qtoptopon 17397 . . . . 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 16967 . . . 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 17452 . 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 1625    e. wcel 1686   A.wral 2545    C_ wss 3154   `'ccnv 4690   ran crn 4692   "cima 4694    Fn wfn 5252   -->wf 5253   -1-1->wf1 5254   -onto->wfo 5255   -1-1-onto->wf1o 5256   ` cfv 5257  (class class class)co 5860   qTop cqtop 13408  TopOnctopon 16634    Cn ccn 16956    Homeo chmeo 17446
This theorem is referenced by:  t0kq  17511
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-map 6776  df-qtop 13412  df-top 16638  df-topon 16641  df-cn 16959  df-hmeo 17448
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