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Theorem qtopt1 28736
Description: If every equivalence class is closed, then the quotient space is T1 . (Contributed by Thierry Arnoux, 5-Jan-2020.)
Hypotheses
Ref Expression
qtopt1.x  |-  X  = 
U. J
qtopt1.1  |-  ( ph  ->  J  e.  Fre )
qtopt1.2  |-  ( ph  ->  F : X -onto-> Y
)
qtopt1.3  |-  ( (
ph  /\  x  e.  Y )  ->  ( `' F " { x } )  e.  (
Clsd `  J )
)
Assertion
Ref Expression
qtopt1  |-  ( ph  ->  ( J qTop  F )  e.  Fre )
Distinct variable groups:    x, F    x, J    ph, x
Allowed substitution hints:    X( x)    Y( x)

Proof of Theorem qtopt1
StepHypRef Expression
1 qtopt1.1 . . . 4  |-  ( ph  ->  J  e.  Fre )
2 t1top 20423 . . . 4  |-  ( J  e.  Fre  ->  J  e.  Top )
31, 2syl 17 . . 3  |-  ( ph  ->  J  e.  Top )
4 qtopt1.2 . . . 4  |-  ( ph  ->  F : X -onto-> Y
)
5 fofn 5808 . . . 4  |-  ( F : X -onto-> Y  ->  F  Fn  X )
64, 5syl 17 . . 3  |-  ( ph  ->  F  Fn  X )
7 qtopt1.x . . . 4  |-  X  = 
U. J
87qtoptop 20792 . . 3  |-  ( ( J  e.  Top  /\  F  Fn  X )  ->  ( J qTop  F )  e.  Top )
93, 6, 8syl2anc 673 . 2  |-  ( ph  ->  ( J qTop  F )  e.  Top )
10 simpr 468 . . . . . 6  |-  ( (
ph  /\  x  e.  U. ( J qTop  F ) )  ->  x  e.  U. ( J qTop  F ) )
117qtopuni 20794 . . . . . . . 8  |-  ( ( J  e.  Top  /\  F : X -onto-> Y )  ->  Y  =  U. ( J qTop  F )
)
123, 4, 11syl2anc 673 . . . . . . 7  |-  ( ph  ->  Y  =  U. ( J qTop  F ) )
1312adantr 472 . . . . . 6  |-  ( (
ph  /\  x  e.  U. ( J qTop  F ) )  ->  Y  =  U. ( J qTop  F ) )
1410, 13eleqtrrd 2552 . . . . 5  |-  ( (
ph  /\  x  e.  U. ( J qTop  F ) )  ->  x  e.  Y )
1514snssd 4108 . . . 4  |-  ( (
ph  /\  x  e.  U. ( J qTop  F ) )  ->  { x }  C_  Y )
16 qtopt1.3 . . . . 5  |-  ( (
ph  /\  x  e.  Y )  ->  ( `' F " { x } )  e.  (
Clsd `  J )
)
1714, 16syldan 478 . . . 4  |-  ( (
ph  /\  x  e.  U. ( J qTop  F ) )  ->  ( `' F " { x }
)  e.  ( Clsd `  J ) )
183, 7jctir 547 . . . . . . 7  |-  ( ph  ->  ( J  e.  Top  /\  X  =  U. J
) )
19 istopon 20017 . . . . . . 7  |-  ( J  e.  (TopOn `  X
)  <->  ( J  e. 
Top  /\  X  =  U. J ) )
2018, 19sylibr 217 . . . . . 6  |-  ( ph  ->  J  e.  (TopOn `  X ) )
21 qtopcld 20805 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  F : X -onto-> Y )  ->  ( { x }  e.  ( Clsd `  ( J qTop  F ) )  <->  ( {
x }  C_  Y  /\  ( `' F " { x } )  e.  ( Clsd `  J
) ) ) )
2220, 4, 21syl2anc 673 . . . . 5  |-  ( ph  ->  ( { x }  e.  ( Clsd `  ( J qTop  F ) )  <->  ( {
x }  C_  Y  /\  ( `' F " { x } )  e.  ( Clsd `  J
) ) ) )
2322adantr 472 . . . 4  |-  ( (
ph  /\  x  e.  U. ( J qTop  F ) )  ->  ( {
x }  e.  (
Clsd `  ( J qTop  F ) )  <->  ( {
x }  C_  Y  /\  ( `' F " { x } )  e.  ( Clsd `  J
) ) ) )
2415, 17, 23mpbir2and 936 . . 3  |-  ( (
ph  /\  x  e.  U. ( J qTop  F ) )  ->  { x }  e.  ( Clsd `  ( J qTop  F ) ) )
2524ralrimiva 2809 . 2  |-  ( ph  ->  A. x  e.  U. ( J qTop  F ) { x }  e.  ( Clsd `  ( J qTop  F ) ) )
26 eqid 2471 . . 3  |-  U. ( J qTop  F )  =  U. ( J qTop  F )
2726ist1 20414 . 2  |-  ( ( J qTop  F )  e. 
Fre 
<->  ( ( J qTop  F
)  e.  Top  /\  A. x  e.  U. ( J qTop  F ) { x }  e.  ( Clsd `  ( J qTop  F ) ) ) )
289, 25, 27sylanbrc 677 1  |-  ( ph  ->  ( J qTop  F )  e.  Fre )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   A.wral 2756    C_ wss 3390   {csn 3959   U.cuni 4190   `'ccnv 4838   "cima 4842    Fn wfn 5584   -onto->wfo 5587   ` cfv 5589  (class class class)co 6308   qTop cqtop 15479   Topctop 19994  TopOnctopon 19995   Clsdccld 20108   Frect1 20400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-qtop 15484  df-top 19998  df-topon 20000  df-cld 20111  df-t1 20407
This theorem is referenced by: (None)
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