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Theorem indistpsx 22376
Description: The indiscrete topology on a set 𝐴 expressed as a topological space, using explicit structure component references. Compare with indistps 22377 and indistps2 22378. The advantage of this version is that the actual function for the structure is evident, and df-ndx 17071 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 17089 and df-tset 17157 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 22377 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 17097 . . . . 5 (Base‘ndx) = 1
32opeq1i 4834 . . . 4 ⟨(Base‘ndx), 𝐴⟩ = ⟨1, 𝐴
4 tsetndx 17238 . . . . 5 (TopSet‘ndx) = 9
54opeq1i 4834 . . . 4 ⟨(TopSet‘ndx), {∅, 𝐴}⟩ = ⟨9, {∅, 𝐴}⟩
63, 5preq12i 4700 . . 3 {⟨(Base‘ndx), 𝐴⟩, ⟨(TopSet‘ndx), {∅, 𝐴}⟩} = {⟨1, 𝐴⟩, ⟨9, {∅, 𝐴}⟩}
71, 6eqtr4i 2764 . 2 𝐾 = {⟨(Base‘ndx), 𝐴⟩, ⟨(TopSet‘ndx), {∅, 𝐴}⟩}
8 indistpsx.a . . . 4 𝐴 ∈ V
9 indistopon 22367 . . . 4 (𝐴 ∈ V → {∅, 𝐴} ∈ (TopOn‘𝐴))
108, 9ax-mp 5 . . 3 {∅, 𝐴} ∈ (TopOn‘𝐴)
1110toponunii 22281 . 2 𝐴 = {∅, 𝐴}
12 indistop 22368 . 2 {∅, 𝐴} ∈ Top
137, 11, 12eltpsi 22310 1 𝐾 ∈ TopSp
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wcel 2107  Vcvv 3444  c0 4283  {cpr 4589  cop 4593  cfv 6497  1c1 11057  9c9 12220  ndxcnx 17070  Basecbs 17088  TopSetcts 17144  TopOnctopon 22275  TopSpctps 22297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-er 8651  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-nn 12159  df-2 12221  df-3 12222  df-4 12223  df-5 12224  df-6 12225  df-7 12226  df-8 12227  df-9 12228  df-n0 12419  df-z 12505  df-uz 12769  df-fz 13431  df-struct 17024  df-slot 17059  df-ndx 17071  df-base 17089  df-tset 17157  df-rest 17309  df-topn 17310  df-top 22259  df-topon 22276  df-topsp 22298
This theorem is referenced by: (None)
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