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Theorem ist0-4 17761
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 17756 . . . . 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 1154 . . . 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 309 . . 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 2745 . 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 17757 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  F  Fn  X )
7 dffn2 5592 . . . 4  |-  ( F  Fn  X  <->  F : X
--> _V )
86, 7sylib 189 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  F : X
--> _V )
9 dff13 6004 . . . 4  |-  ( F : X -1-1-> _V  <->  ( F : X --> _V  /\  A. z  e.  X  A. w  e.  X  ( ( F `  z )  =  ( F `  w )  ->  z  =  w ) ) )
109baib 872 . . 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 17408 . 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 278 1  |-  ( J  e.  (TopOn `  X
)  ->  ( J  e.  Kol2  <->  F : X -1-1-> _V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   {crab 2709   _Vcvv 2956    e. cmpt 4266    Fn wfn 5449   -->wf 5450   -1-1->wf1 5451   ` cfv 5454  TopOnctopon 16959   Kol2ct0 17370
This theorem is referenced by:  t0kq  17850
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fv 5462  df-topon 16966  df-t0 17377
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