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Theorem ist0-4 17436
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 17431 . . . . 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 1152 . . . 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 308 . . 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 2596 . 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 17432 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  F  Fn  X )
7 dffn2 5406 . . . 4  |-  ( F  Fn  X  <->  F : X
--> _V )
86, 7sylib 188 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  F : X
--> _V )
9 dff13 5799 . . . 4  |-  ( F : X -1-1-> _V  <->  ( F : X --> _V  /\  A. z  e.  X  A. w  e.  X  ( ( F `  z )  =  ( F `  w )  ->  z  =  w ) ) )
109baib 871 . . 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 15 . 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 17088 . 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 277 1  |-  ( J  e.  (TopOn `  X
)  ->  ( J  e.  Kol2  <->  F : X -1-1-> _V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560   _Vcvv 2801    e. cmpt 4093    Fn wfn 5266   -->wf 5267   -1-1->wf1 5268   ` cfv 5271  TopOnctopon 16648   Kol2ct0 17050
This theorem is referenced by:  t0kq  17525
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fv 5279  df-topon 16655  df-t0 17057
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