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Theorem qtopf1 17853
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 5643 . . . 4  |-  ( F : X -1-1-> Y  ->  F  Fn  X )
42, 3syl 16 . . 3  |-  ( ph  ->  F  Fn  X )
5 qtopid 17742 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  F  Fn  X )  ->  F  e.  ( J  Cn  ( J qTop  F ) ) )
61, 4, 5syl2anc 644 . 2  |-  ( ph  ->  F  e.  ( J  Cn  ( J qTop  F
) ) )
7 f1f1orn 5688 . . . 4  |-  ( F : X -1-1-> Y  ->  F : X -1-1-onto-> ran  F )
8 f1ocnv 5690 . . . 4  |-  ( F : X -1-1-onto-> ran  F  ->  `' F : ran  F -1-1-onto-> X )
9 f1of 5677 . . . 4  |-  ( `' F : ran  F -1-1-onto-> X  ->  `' F : ran  F --> X )
102, 7, 8, 94syl 20 . . 3  |-  ( ph  ->  `' F : ran  F --> X )
11 imacnvcnv 5337 . . . . 5  |-  ( `' `' F " x )  =  ( F "
x )
12 imassrn 5219 . . . . . . 7  |-  ( F
" x )  C_  ran  F
1312a1i 11 . . . . . 6  |-  ( (
ph  /\  x  e.  J )  ->  ( F " x )  C_  ran  F )
142adantr 453 . . . . . . . 8  |-  ( (
ph  /\  x  e.  J )  ->  F : X -1-1-> Y )
15 toponss 16999 . . . . . . . . 9  |-  ( ( J  e.  (TopOn `  X )  /\  x  e.  J )  ->  x  C_  X )
161, 15sylan 459 . . . . . . . 8  |-  ( (
ph  /\  x  e.  J )  ->  x  C_  X )
17 f1imacnv 5694 . . . . . . . 8  |-  ( ( F : X -1-1-> Y  /\  x  C_  X )  ->  ( `' F " ( F " x
) )  =  x )
1814, 16, 17syl2anc 644 . . . . . . 7  |-  ( (
ph  /\  x  e.  J )  ->  ( `' F " ( F
" x ) )  =  x )
19 simpr 449 . . . . . . 7  |-  ( (
ph  /\  x  e.  J )  ->  x  e.  J )
2018, 19eqeltrd 2512 . . . . . 6  |-  ( (
ph  /\  x  e.  J )  ->  ( `' F " ( F
" x ) )  e.  J )
21 dffn4 5662 . . . . . . . . 9  |-  ( F  Fn  X  <->  F : X -onto-> ran  F )
224, 21sylib 190 . . . . . . . 8  |-  ( ph  ->  F : X -onto-> ran  F )
23 elqtop3 17740 . . . . . . . 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 ) ) )
241, 22, 23syl2anc 644 . . . . . . 7  |-  ( ph  ->  ( ( F "
x )  e.  ( J qTop  F )  <->  ( ( F " x )  C_  ran  F  /\  ( `' F " ( F
" x ) )  e.  J ) ) )
2524adantr 453 . . . . . 6  |-  ( (
ph  /\  x  e.  J )  ->  (
( F " x
)  e.  ( J qTop 
F )  <->  ( ( F " x )  C_  ran  F  /\  ( `' F " ( F
" x ) )  e.  J ) ) )
2613, 20, 25mpbir2and 890 . . . . 5  |-  ( (
ph  /\  x  e.  J )  ->  ( F " x )  e.  ( J qTop  F ) )
2711, 26syl5eqel 2522 . . . 4  |-  ( (
ph  /\  x  e.  J )  ->  ( `' `' F " x )  e.  ( J qTop  F
) )
2827ralrimiva 2791 . . 3  |-  ( ph  ->  A. x  e.  J  ( `' `' F " x )  e.  ( J qTop  F
) )
29 qtoptopon 17741 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  F : X -onto-> ran  F )  -> 
( J qTop  F )  e.  (TopOn `  ran  F ) )
301, 22, 29syl2anc 644 . . . 4  |-  ( ph  ->  ( J qTop  F )  e.  (TopOn `  ran  F ) )
31 iscn 17304 . . . 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
) ) ) )
3230, 1, 31syl2anc 644 . . 3  |-  ( ph  ->  ( `' F  e.  ( ( J qTop  F
)  Cn  J )  <-> 
( `' F : ran  F --> X  /\  A. x  e.  J  ( `' `' F " x )  e.  ( J qTop  F
) ) ) )
3310, 28, 32mpbir2and 890 . 2  |-  ( ph  ->  `' F  e.  (
( J qTop  F )  Cn  J ) )
34 ishmeo 17796 . 2  |-  ( F  e.  ( J  Homeo  ( J qTop  F ) )  <-> 
( F  e.  ( J  Cn  ( J qTop 
F ) )  /\  `' F  e.  (
( J qTop  F )  Cn  J ) ) )
356, 33, 34sylanbrc 647 1  |-  ( ph  ->  F  e.  ( J 
Homeo  ( J qTop  F ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707    C_ wss 3322   `'ccnv 4880   ran crn 4882   "cima 4884    Fn wfn 5452   -->wf 5453   -1-1->wf1 5454   -onto->wfo 5455   -1-1-onto->wf1o 5456   ` cfv 5457  (class class class)co 6084   qTop cqtop 13734  TopOnctopon 16964    Cn ccn 17293    Homeo chmeo 17790
This theorem is referenced by:  t0kq  17855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-map 7023  df-qtop 13738  df-top 16968  df-topon 16971  df-cn 17296  df-hmeo 17792
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