Theorem ist0-4 21532

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.

Step	Hyp	Ref	Expression
1		kqval.2	. . . . . 6 |- F = ( x e. X |-> { y e. J | x e. y } )
2	1	kqfeq 21527	. . . . 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 ) ) )
3	2	3expb 1266	. . . 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 ) ) )
4	3	imbi1d 331	. . 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 ) ) )
5	4	2ralbidva 2988	. 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 ) ) )
6	1	kqffn 21528	. . . 4 |- ( J e. (TopOn ` X ) -> F Fn X )
7		dffn2 6047	. . . 4 |- ( F Fn X <-> F : X --> _V )
8	6, 7	sylib 208	. . 3 |- ( J e. (TopOn ` X ) -> F : X --> _V )
9		dff13 6512	. . . 4 |- ( F : X -1-1-> _V <-> ( F : X --> _V /\ A. z e. X A. w e. X ( ( F ` z ) = ( F ` w ) -> z = w ) ) )
10	9	baib 944	. . 3 |- ( F : X --> _V -> ( F : X -1-1-> _V <-> A. z e. X A. w e. X ( ( F ` z ) = ( F ` w ) -> z = w ) ) )
11	8, 10	syl 17	. 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 21148	. 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 ) ) )
13	5, 11, 12	3bitr4rd 301	1 |- ( J e. (TopOn ` X ) -> ( J e. Kol2 <-> F : X -1-1-> _V ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   |-> cmpt 4729   Fn wfn 5883   -->wf 5884   -1-1->wf1 5885   ` cfv 5888  TopOnctopon 20715   Kol2ct0 21110

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fv 5896  df-topon 20716  df-t0 21117

This theorem is referenced by:  t0kq  21621