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Theorem ist0-4 19993
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 19988 . . . . 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 1197 . . . 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 2906 . 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 19989 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  F  Fn  X )
7 dffn2 5732 . . . 4  |-  ( F  Fn  X  <->  F : X
--> _V )
86, 7sylib 196 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  F : X
--> _V )
9 dff13 6154 . . . 4  |-  ( F : X -1-1-> _V  <->  ( F : X --> _V  /\  A. z  e.  X  A. w  e.  X  ( ( F `  z )  =  ( F `  w )  ->  z  =  w ) ) )
109baib 901 . . 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 19639 . 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 1379    e. wcel 1767   A.wral 2814   {crab 2818   _Vcvv 3113    |-> cmpt 4505    Fn wfn 5583   -->wf 5584   -1-1->wf1 5585   ` cfv 5588  TopOnctopon 19190   Kol2ct0 19601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fv 5596  df-topon 19197  df-t0 19608
This theorem is referenced by:  t0kq  20082
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