![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > eltpsg | Structured version Visualization version GIF version |
Description: Properties that determine a topological space from a construction (using no explicit indices). (Contributed by Mario Carneiro, 13-Aug-2015.) (Revised by AV, 31-Oct-2024.) |
Ref | Expression |
---|---|
eltpsi.k | ⊢ 𝐾 = {⟨(Base‘ndx), 𝐴⟩, ⟨(TopSet‘ndx), 𝐽⟩} |
Ref | Expression |
---|---|
eltpsg | ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐾 ∈ TopSp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eltpsi.k | . . . . 5 ⊢ 𝐾 = {⟨(Base‘ndx), 𝐴⟩, ⟨(TopSet‘ndx), 𝐽⟩} | |
2 | basendxlttsetndx 17307 | . . . . 5 ⊢ (Base‘ndx) < (TopSet‘ndx) | |
3 | tsetndxnn 17306 | . . . . 5 ⊢ (TopSet‘ndx) ∈ ℕ | |
4 | tsetid 17305 | . . . . 5 ⊢ TopSet = Slot (TopSet‘ndx) | |
5 | 1, 2, 3, 4 | 2strop1 17179 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐽 = (TopSet‘𝐾)) |
6 | toponmax 22779 | . . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐴 ∈ 𝐽) | |
7 | 1, 2, 3 | 2strbas1 17178 | . . . . . 6 ⊢ (𝐴 ∈ 𝐽 → 𝐴 = (Base‘𝐾)) |
8 | 6, 7 | syl 17 | . . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐴 = (Base‘𝐾)) |
9 | 8 | fveq2d 6888 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝐴) → (TopOn‘𝐴) = (TopOn‘(Base‘𝐾))) |
10 | 5, 9 | eleq12d 2821 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝐴) → (𝐽 ∈ (TopOn‘𝐴) ↔ (TopSet‘𝐾) ∈ (TopOn‘(Base‘𝐾)))) |
11 | 10 | ibi 267 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝐴) → (TopSet‘𝐾) ∈ (TopOn‘(Base‘𝐾))) |
12 | eqid 2726 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
13 | eqid 2726 | . . 3 ⊢ (TopSet‘𝐾) = (TopSet‘𝐾) | |
14 | 12, 13 | tsettps 22794 | . 2 ⊢ ((TopSet‘𝐾) ∈ (TopOn‘(Base‘𝐾)) → 𝐾 ∈ TopSp) |
15 | 11, 14 | syl 17 | 1 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐾 ∈ TopSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 {cpr 4625 ⟨cop 4629 ‘cfv 6536 ndxcnx 17133 Basecbs 17151 TopSetcts 17210 TopOnctopon 22763 TopSpctps 22785 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-uz 12824 df-fz 13488 df-struct 17087 df-slot 17122 df-ndx 17134 df-base 17152 df-tset 17223 df-rest 17375 df-topn 17376 df-top 22747 df-topon 22764 df-topsp 22786 |
This theorem is referenced by: eltpsi 22798 stoig 23018 |
Copyright terms: Public domain | W3C validator |