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Theorem ist0-4 19300
Description: The topological indistinguishability map is injective iff the space is T0. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
kqval.2  |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )
Assertion
Ref Expression
ist0-4  |-  ( J  e.  (TopOn `  X
)  ->  ( J  e.  Kol2  <->  F : X -1-1-> _V ) )
Distinct variable groups:    x, y, J    x, X, y
Allowed substitution hints:    F( x, y)

Proof of Theorem ist0-4
Dummy variables  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 kqval.2 . . . . . 6  |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )
21kqfeq 19295 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  z  e.  X  /\  w  e.  X )  ->  (
( F `  z
)  =  ( F `
 w )  <->  A. y  e.  J  ( z  e.  y  <->  w  e.  y
) ) )
323expb 1188 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  (
z  e.  X  /\  w  e.  X )
)  ->  ( ( F `  z )  =  ( F `  w )  <->  A. y  e.  J  ( z  e.  y  <->  w  e.  y
) ) )
43imbi1d 317 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  (
z  e.  X  /\  w  e.  X )
)  ->  ( (
( F `  z
)  =  ( F `
 w )  -> 
z  =  w )  <-> 
( A. y  e.  J  ( z  e.  y  <->  w  e.  y
)  ->  z  =  w ) ) )
542ralbidva 2753 . 2  |-  ( J  e.  (TopOn `  X
)  ->  ( A. z  e.  X  A. w  e.  X  (
( F `  z
)  =  ( F `
 w )  -> 
z  =  w )  <->  A. z  e.  X  A. w  e.  X  ( A. y  e.  J  ( z  e.  y  <-> 
w  e.  y )  ->  z  =  w ) ) )
61kqffn 19296 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  F  Fn  X )
7 dffn2 5558 . . . 4  |-  ( F  Fn  X  <->  F : X
--> _V )
86, 7sylib 196 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  F : X
--> _V )
9 dff13 5969 . . . 4  |-  ( F : X -1-1-> _V  <->  ( F : X --> _V  /\  A. z  e.  X  A. w  e.  X  ( ( F `  z )  =  ( F `  w )  ->  z  =  w ) ) )
109baib 896 . . 3  |-  ( F : X --> _V  ->  ( F : X -1-1-> _V  <->  A. z  e.  X  A. w  e.  X  (
( F `  z
)  =  ( F `
 w )  -> 
z  =  w ) ) )
118, 10syl 16 . 2  |-  ( J  e.  (TopOn `  X
)  ->  ( F : X -1-1-> _V  <->  A. z  e.  X  A. w  e.  X  ( ( F `  z )  =  ( F `  w )  ->  z  =  w ) ) )
12 ist0-2 18946 . 2  |-  ( J  e.  (TopOn `  X
)  ->  ( J  e.  Kol2  <->  A. z  e.  X  A. w  e.  X  ( A. y  e.  J  ( z  e.  y  <-> 
w  e.  y )  ->  z  =  w ) ) )
135, 11, 123bitr4rd 286 1  |-  ( J  e.  (TopOn `  X
)  ->  ( J  e.  Kol2  <->  F : X -1-1-> _V ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2713   {crab 2717   _Vcvv 2970    e. cmpt 4348    Fn wfn 5411   -->wf 5412   -1-1->wf1 5413   ` cfv 5416  TopOnctopon 18497   Kol2ct0 18908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fv 5424  df-topon 18504  df-t0 18915
This theorem is referenced by:  t0kq  19389
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