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Theorem qtopf1 17809
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 5607 . . . 4  |-  ( F : X -1-1-> Y  ->  F  Fn  X )
42, 3syl 16 . . 3  |-  ( ph  ->  F  Fn  X )
5 qtopid 17698 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  F  Fn  X )  ->  F  e.  ( J  Cn  ( J qTop  F ) ) )
61, 4, 5syl2anc 643 . 2  |-  ( ph  ->  F  e.  ( J  Cn  ( J qTop  F
) ) )
7 f1f1orn 5652 . . . . 5  |-  ( F : X -1-1-> Y  ->  F : X -1-1-onto-> ran  F )
82, 7syl 16 . . . 4  |-  ( ph  ->  F : X -1-1-onto-> ran  F
)
9 f1ocnv 5654 . . . 4  |-  ( F : X -1-1-onto-> ran  F  ->  `' F : ran  F -1-1-onto-> X )
10 f1of 5641 . . . 4  |-  ( `' F : ran  F -1-1-onto-> X  ->  `' F : ran  F --> X )
118, 9, 103syl 19 . . 3  |-  ( ph  ->  `' F : ran  F --> X )
12 imacnvcnv 5301 . . . . 5  |-  ( `' `' F " x )  =  ( F "
x )
13 imassrn 5183 . . . . . . 7  |-  ( F
" x )  C_  ran  F
1413a1i 11 . . . . . 6  |-  ( (
ph  /\  x  e.  J )  ->  ( F " x )  C_  ran  F )
152adantr 452 . . . . . . . 8  |-  ( (
ph  /\  x  e.  J )  ->  F : X -1-1-> Y )
16 toponss 16957 . . . . . . . . 9  |-  ( ( J  e.  (TopOn `  X )  /\  x  e.  J )  ->  x  C_  X )
171, 16sylan 458 . . . . . . . 8  |-  ( (
ph  /\  x  e.  J )  ->  x  C_  X )
18 f1imacnv 5658 . . . . . . . 8  |-  ( ( F : X -1-1-> Y  /\  x  C_  X )  ->  ( `' F " ( F " x
) )  =  x )
1915, 17, 18syl2anc 643 . . . . . . 7  |-  ( (
ph  /\  x  e.  J )  ->  ( `' F " ( F
" x ) )  =  x )
20 simpr 448 . . . . . . 7  |-  ( (
ph  /\  x  e.  J )  ->  x  e.  J )
2119, 20eqeltrd 2486 . . . . . 6  |-  ( (
ph  /\  x  e.  J )  ->  ( `' F " ( F
" x ) )  e.  J )
22 dffn4 5626 . . . . . . . . 9  |-  ( F  Fn  X  <->  F : X -onto-> ran  F )
234, 22sylib 189 . . . . . . . 8  |-  ( ph  ->  F : X -onto-> ran  F )
24 elqtop3 17696 . . . . . . . 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 643 . . . . . . 7  |-  ( ph  ->  ( ( F "
x )  e.  ( J qTop  F )  <->  ( ( F " x )  C_  ran  F  /\  ( `' F " ( F
" x ) )  e.  J ) ) )
2625adantr 452 . . . . . 6  |-  ( (
ph  /\  x  e.  J )  ->  (
( F " x
)  e.  ( J qTop 
F )  <->  ( ( F " x )  C_  ran  F  /\  ( `' F " ( F
" x ) )  e.  J ) ) )
2714, 21, 26mpbir2and 889 . . . . 5  |-  ( (
ph  /\  x  e.  J )  ->  ( F " x )  e.  ( J qTop  F ) )
2812, 27syl5eqel 2496 . . . 4  |-  ( (
ph  /\  x  e.  J )  ->  ( `' `' F " x )  e.  ( J qTop  F
) )
2928ralrimiva 2757 . . 3  |-  ( ph  ->  A. x  e.  J  ( `' `' F " x )  e.  ( J qTop  F
) )
30 qtoptopon 17697 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  F : X -onto-> ran  F )  -> 
( J qTop  F )  e.  (TopOn `  ran  F ) )
311, 23, 30syl2anc 643 . . . 4  |-  ( ph  ->  ( J qTop  F )  e.  (TopOn `  ran  F ) )
32 iscn 17261 . . . 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 643 . . 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 889 . 2  |-  ( ph  ->  `' F  e.  (
( J qTop  F )  Cn  J ) )
35 ishmeo 17752 . 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 646 1  |-  ( ph  ->  F  e.  ( J 
Homeo  ( J qTop  F ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2674    C_ wss 3288   `'ccnv 4844   ran crn 4846   "cima 4848    Fn wfn 5416   -->wf 5417   -1-1->wf1 5418   -onto->wfo 5419   -1-1-onto->wf1o 5420   ` cfv 5421  (class class class)co 6048   qTop cqtop 13692  TopOnctopon 16922    Cn ccn 17250    Homeo chmeo 17746
This theorem is referenced by:  t0kq  17811
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-map 6987  df-qtop 13696  df-top 16926  df-topon 16929  df-cn 17253  df-hmeo 17748
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