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Theorem indistpsx 21016
Description: The indiscrete topology on a set 𝐴 expressed as a topological space, using explicit structure component references. Compare with indistps 21017 and indistps2 21018. The advantage of this version is that the actual function for the structure is evident, and df-ndx 16062 is not needed, nor any other special definition outside of basic set theory. The disadvantage is that if the indices of the component definitions df-base 16065 and df-tset 16162 are changed in the future, this theorem will also have to be changed. Note: This theorem has hard-coded structure indices for demonstration purposes. It is not intended for general use; use indistps 21017 instead. (New usage is discouraged.) (Contributed by FL, 19-Jul-2006.)
Hypotheses
Ref Expression
indistpsx.a 𝐴 ∈ V
indistpsx.k 𝐾 = {⟨1, 𝐴⟩, ⟨9, {∅, 𝐴}⟩}
Assertion
Ref Expression
indistpsx 𝐾 ∈ TopSp

Proof of Theorem indistpsx
StepHypRef Expression
1 indistpsx.k . . 3 𝐾 = {⟨1, 𝐴⟩, ⟨9, {∅, 𝐴}⟩}
2 basendx 16125 . . . . 5 (Base‘ndx) = 1
32opeq1i 4556 . . . 4 ⟨(Base‘ndx), 𝐴⟩ = ⟨1, 𝐴
4 tsetndx 16242 . . . . 5 (TopSet‘ndx) = 9
54opeq1i 4556 . . . 4 ⟨(TopSet‘ndx), {∅, 𝐴}⟩ = ⟨9, {∅, 𝐴}⟩
63, 5preq12i 4417 . . 3 {⟨(Base‘ndx), 𝐴⟩, ⟨(TopSet‘ndx), {∅, 𝐴}⟩} = {⟨1, 𝐴⟩, ⟨9, {∅, 𝐴}⟩}
71, 6eqtr4i 2785 . 2 𝐾 = {⟨(Base‘ndx), 𝐴⟩, ⟨(TopSet‘ndx), {∅, 𝐴}⟩}
8 indistpsx.a . . . 4 𝐴 ∈ V
9 indistopon 21007 . . . 4 (𝐴 ∈ V → {∅, 𝐴} ∈ (TopOn‘𝐴))
108, 9ax-mp 5 . . 3 {∅, 𝐴} ∈ (TopOn‘𝐴)
1110toponunii 20923 . 2 𝐴 = {∅, 𝐴}
12 indistop 21008 . 2 {∅, 𝐴} ∈ Top
137, 11, 12eltpsi 20950 1 𝐾 ∈ TopSp
Colors of variables: wff setvar class
Syntax hints:   = wceq 1632  wcel 2139  Vcvv 3340  c0 4058  {cpr 4323  cop 4327  cfv 6049  1c1 10129  9c9 11269  ndxcnx 16056  Basecbs 16059  TopSetcts 16149  TopOnctopon 20917  TopSpctps 20938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-er 7911  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-3 11272  df-4 11273  df-5 11274  df-6 11275  df-7 11276  df-8 11277  df-9 11278  df-n0 11485  df-z 11570  df-uz 11880  df-fz 12520  df-struct 16061  df-ndx 16062  df-slot 16063  df-base 16065  df-tset 16162  df-rest 16285  df-topn 16286  df-top 20901  df-topon 20918  df-topsp 20939
This theorem is referenced by: (None)
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