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Theorem qtopf1 20052
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 5780 . . . 4  |-  ( F : X -1-1-> Y  ->  F  Fn  X )
42, 3syl 16 . . 3  |-  ( ph  ->  F  Fn  X )
5 qtopid 19941 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  F  Fn  X )  ->  F  e.  ( J  Cn  ( J qTop  F ) ) )
61, 4, 5syl2anc 661 . 2  |-  ( ph  ->  F  e.  ( J  Cn  ( J qTop  F
) ) )
7 f1f1orn 5825 . . . 4  |-  ( F : X -1-1-> Y  ->  F : X -1-1-onto-> ran  F )
8 f1ocnv 5826 . . . 4  |-  ( F : X -1-1-onto-> ran  F  ->  `' F : ran  F -1-1-onto-> X )
9 f1of 5814 . . . 4  |-  ( `' F : ran  F -1-1-onto-> X  ->  `' F : ran  F --> X )
102, 7, 8, 94syl 21 . . 3  |-  ( ph  ->  `' F : ran  F --> X )
11 imacnvcnv 5470 . . . . 5  |-  ( `' `' F " x )  =  ( F "
x )
12 imassrn 5346 . . . . . . 7  |-  ( F
" x )  C_  ran  F
1312a1i 11 . . . . . 6  |-  ( (
ph  /\  x  e.  J )  ->  ( F " x )  C_  ran  F )
142adantr 465 . . . . . . . 8  |-  ( (
ph  /\  x  e.  J )  ->  F : X -1-1-> Y )
15 toponss 19197 . . . . . . . . 9  |-  ( ( J  e.  (TopOn `  X )  /\  x  e.  J )  ->  x  C_  X )
161, 15sylan 471 . . . . . . . 8  |-  ( (
ph  /\  x  e.  J )  ->  x  C_  X )
17 f1imacnv 5830 . . . . . . . 8  |-  ( ( F : X -1-1-> Y  /\  x  C_  X )  ->  ( `' F " ( F " x
) )  =  x )
1814, 16, 17syl2anc 661 . . . . . . 7  |-  ( (
ph  /\  x  e.  J )  ->  ( `' F " ( F
" x ) )  =  x )
19 simpr 461 . . . . . . 7  |-  ( (
ph  /\  x  e.  J )  ->  x  e.  J )
2018, 19eqeltrd 2555 . . . . . 6  |-  ( (
ph  /\  x  e.  J )  ->  ( `' F " ( F
" x ) )  e.  J )
21 dffn4 5799 . . . . . . . . 9  |-  ( F  Fn  X  <->  F : X -onto-> ran  F )
224, 21sylib 196 . . . . . . . 8  |-  ( ph  ->  F : X -onto-> ran  F )
23 elqtop3 19939 . . . . . . . 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 661 . . . . . . 7  |-  ( ph  ->  ( ( F "
x )  e.  ( J qTop  F )  <->  ( ( F " x )  C_  ran  F  /\  ( `' F " ( F
" x ) )  e.  J ) ) )
2524adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  J )  ->  (
( F " x
)  e.  ( J qTop 
F )  <->  ( ( F " x )  C_  ran  F  /\  ( `' F " ( F
" x ) )  e.  J ) ) )
2613, 20, 25mpbir2and 920 . . . . 5  |-  ( (
ph  /\  x  e.  J )  ->  ( F " x )  e.  ( J qTop  F ) )
2711, 26syl5eqel 2559 . . . 4  |-  ( (
ph  /\  x  e.  J )  ->  ( `' `' F " x )  e.  ( J qTop  F
) )
2827ralrimiva 2878 . . 3  |-  ( ph  ->  A. x  e.  J  ( `' `' F " x )  e.  ( J qTop  F
) )
29 qtoptopon 19940 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  F : X -onto-> ran  F )  -> 
( J qTop  F )  e.  (TopOn `  ran  F ) )
301, 22, 29syl2anc 661 . . . 4  |-  ( ph  ->  ( J qTop  F )  e.  (TopOn `  ran  F ) )
31 iscn 19502 . . . 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 661 . . 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 920 . 2  |-  ( ph  ->  `' F  e.  (
( J qTop  F )  Cn  J ) )
34 ishmeo 19995 . 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 664 1  |-  ( ph  ->  F  e.  ( J
Homeo ( J qTop  F ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814    C_ wss 3476   `'ccnv 4998   ran crn 5000   "cima 5002    Fn wfn 5581   -->wf 5582   -1-1->wf1 5583   -onto->wfo 5584   -1-1-onto->wf1o 5585   ` cfv 5586  (class class class)co 6282   qTop cqtop 14754  TopOnctopon 19162    Cn ccn 19491   Homeochmeo 19989
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 6574
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 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-map 7419  df-qtop 14758  df-top 19166  df-topon 19169  df-cn 19494  df-hmeo 19991
This theorem is referenced by:  t0kq  20054
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