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Mirrors > Home > MPE Home > Th. List > indistpsx | Structured version Visualization version GIF version |
Description: The indiscrete topology on a set 𝐴 expressed as a topological space, using explicit structure component references. Compare with indistps 22932 and indistps2 22933. The advantage of this version is that the actual function for the structure is evident, and df-ndx 17162 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 17180 and df-tset 17251 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 22932 instead. (New usage is discouraged.) (Contributed by FL, 19-Jul-2006.) |
Ref | Expression |
---|---|
indistpsx.a | ⊢ 𝐴 ∈ V |
indistpsx.k | ⊢ 𝐾 = {⟨1, 𝐴⟩, ⟨9, {∅, 𝐴}⟩} |
Ref | Expression |
---|---|
indistpsx | ⊢ 𝐾 ∈ TopSp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | indistpsx.k | . . 3 ⊢ 𝐾 = {⟨1, 𝐴⟩, ⟨9, {∅, 𝐴}⟩} | |
2 | basendx 17188 | . . . . 5 ⊢ (Base‘ndx) = 1 | |
3 | 2 | opeq1i 4872 | . . . 4 ⊢ ⟨(Base‘ndx), 𝐴⟩ = ⟨1, 𝐴⟩ |
4 | tsetndx 17332 | . . . . 5 ⊢ (TopSet‘ndx) = 9 | |
5 | 4 | opeq1i 4872 | . . . 4 ⊢ ⟨(TopSet‘ndx), {∅, 𝐴}⟩ = ⟨9, {∅, 𝐴}⟩ |
6 | 3, 5 | preq12i 4738 | . . 3 ⊢ {⟨(Base‘ndx), 𝐴⟩, ⟨(TopSet‘ndx), {∅, 𝐴}⟩} = {⟨1, 𝐴⟩, ⟨9, {∅, 𝐴}⟩} |
7 | 1, 6 | eqtr4i 2756 | . 2 ⊢ 𝐾 = {⟨(Base‘ndx), 𝐴⟩, ⟨(TopSet‘ndx), {∅, 𝐴}⟩} |
8 | indistpsx.a | . . . 4 ⊢ 𝐴 ∈ V | |
9 | indistopon 22922 | . . . 4 ⊢ (𝐴 ∈ V → {∅, 𝐴} ∈ (TopOn‘𝐴)) | |
10 | 8, 9 | ax-mp 5 | . . 3 ⊢ {∅, 𝐴} ∈ (TopOn‘𝐴) |
11 | 10 | toponunii 22836 | . 2 ⊢ 𝐴 = ∪ {∅, 𝐴} |
12 | indistop 22923 | . 2 ⊢ {∅, 𝐴} ∈ Top | |
13 | 7, 11, 12 | eltpsi 22865 | 1 ⊢ 𝐾 ∈ TopSp |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 Vcvv 3463 ∅c0 4318 {cpr 4626 ⟨cop 4630 ‘cfv 6543 1c1 11139 9c9 12304 ndxcnx 17161 Basecbs 17179 TopSetcts 17238 TopOnctopon 22830 TopSpctps 22852 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-uz 12853 df-fz 13517 df-struct 17115 df-slot 17150 df-ndx 17162 df-base 17180 df-tset 17251 df-rest 17403 df-topn 17404 df-top 22814 df-topon 22831 df-topsp 22853 |
This theorem is referenced by: (None) |
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