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Theorem qtopt1 28662
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 20346 . . . 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 5795 . . . 4  |-  ( F : X -onto-> Y  ->  F  Fn  X )
64, 5syl 17 . . 3  |-  ( ph  ->  F  Fn  X )
7 qtopt1.x . . . 4  |-  X  = 
U. J
87qtoptop 20715 . . 3  |-  ( ( J  e.  Top  /\  F  Fn  X )  ->  ( J qTop  F )  e.  Top )
93, 6, 8syl2anc 667 . 2  |-  ( ph  ->  ( J qTop  F )  e.  Top )
10 simpr 463 . . . . . 6  |-  ( (
ph  /\  x  e.  U. ( J qTop  F ) )  ->  x  e.  U. ( J qTop  F ) )
117qtopuni 20717 . . . . . . . 8  |-  ( ( J  e.  Top  /\  F : X -onto-> Y )  ->  Y  =  U. ( J qTop  F )
)
123, 4, 11syl2anc 667 . . . . . . 7  |-  ( ph  ->  Y  =  U. ( J qTop  F ) )
1312adantr 467 . . . . . 6  |-  ( (
ph  /\  x  e.  U. ( J qTop  F ) )  ->  Y  =  U. ( J qTop  F ) )
1410, 13eleqtrrd 2532 . . . . 5  |-  ( (
ph  /\  x  e.  U. ( J qTop  F ) )  ->  x  e.  Y )
1514snssd 4117 . . . 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 473 . . . 4  |-  ( (
ph  /\  x  e.  U. ( J qTop  F ) )  ->  ( `' F " { x }
)  e.  ( Clsd `  J ) )
183, 7jctir 541 . . . . . . 7  |-  ( ph  ->  ( J  e.  Top  /\  X  =  U. J
) )
19 istopon 19940 . . . . . . 7  |-  ( J  e.  (TopOn `  X
)  <->  ( J  e. 
Top  /\  X  =  U. J ) )
2018, 19sylibr 216 . . . . . 6  |-  ( ph  ->  J  e.  (TopOn `  X ) )
21 qtopcld 20728 . . . . . 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 667 . . . . 5  |-  ( ph  ->  ( { x }  e.  ( Clsd `  ( J qTop  F ) )  <->  ( {
x }  C_  Y  /\  ( `' F " { x } )  e.  ( Clsd `  J
) ) ) )
2322adantr 467 . . . 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 933 . . 3  |-  ( (
ph  /\  x  e.  U. ( J qTop  F ) )  ->  { x }  e.  ( Clsd `  ( J qTop  F ) ) )
2524ralrimiva 2802 . 2  |-  ( ph  ->  A. x  e.  U. ( J qTop  F ) { x }  e.  ( Clsd `  ( J qTop  F ) ) )
26 eqid 2451 . . 3  |-  U. ( J qTop  F )  =  U. ( J qTop  F )
2726ist1 20337 . 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 670 1  |-  ( ph  ->  ( J qTop  F )  e.  Fre )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1444    e. wcel 1887   A.wral 2737    C_ wss 3404   {csn 3968   U.cuni 4198   `'ccnv 4833   "cima 4837    Fn wfn 5577   -onto->wfo 5580   ` cfv 5582  (class class class)co 6290   qTop cqtop 15401   Topctop 19917  TopOnctopon 19918   Clsdccld 20031   Frect1 20323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-qtop 15406  df-top 19921  df-topon 19923  df-cld 20034  df-t1 20330
This theorem is referenced by: (None)
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