Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > eltpsg | GIF version |
Description: Properties that determine a topological space from a construction (using no explicit indices). (Contributed by Mario Carneiro, 13-Aug-2015.) |
Ref | Expression |
---|---|
eltpsi.k | ⊢ 𝐾 = {〈(Base‘ndx), 𝐴〉, 〈(TopSet‘ndx), 𝐽〉} |
Ref | Expression |
---|---|
eltpsg | ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐾 ∈ TopSp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | toponmax 12570 | . . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐴 ∈ 𝐽) | |
2 | eltpsi.k | . . . . . 6 ⊢ 𝐾 = {〈(Base‘ndx), 𝐴〉, 〈(TopSet‘ndx), 𝐽〉} | |
3 | df-tset 12418 | . . . . . 6 ⊢ TopSet = Slot 9 | |
4 | 1lt9 9052 | . . . . . 6 ⊢ 1 < 9 | |
5 | 9nn 9016 | . . . . . 6 ⊢ 9 ∈ ℕ | |
6 | 2, 3, 4, 5 | 2stropg 12439 | . . . . 5 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐽 ∈ (TopOn‘𝐴)) → 𝐽 = (TopSet‘𝐾)) |
7 | 1, 6 | mpancom 419 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐽 = (TopSet‘𝐾)) |
8 | 2, 3, 4, 5 | 2strbasg 12438 | . . . . . 6 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐽 ∈ (TopOn‘𝐴)) → 𝐴 = (Base‘𝐾)) |
9 | 1, 8 | mpancom 419 | . . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐴 = (Base‘𝐾)) |
10 | 9 | fveq2d 5484 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝐴) → (TopOn‘𝐴) = (TopOn‘(Base‘𝐾))) |
11 | 7, 10 | eleq12d 2235 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝐴) → (𝐽 ∈ (TopOn‘𝐴) ↔ (TopSet‘𝐾) ∈ (TopOn‘(Base‘𝐾)))) |
12 | 11 | ibi 175 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝐴) → (TopSet‘𝐾) ∈ (TopOn‘(Base‘𝐾))) |
13 | eqid 2164 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
14 | eqid 2164 | . . 3 ⊢ (TopSet‘𝐾) = (TopSet‘𝐾) | |
15 | 13, 14 | tsettps 12583 | . 2 ⊢ ((TopSet‘𝐾) ∈ (TopOn‘(Base‘𝐾)) → 𝐾 ∈ TopSp) |
16 | 12, 15 | syl 14 | 1 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐾 ∈ TopSp) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 ∈ wcel 2135 {cpr 3571 〈cop 3573 ‘cfv 5182 9c9 8906 ndxcnx 12334 Basecbs 12337 TopSetcts 12405 TopOnctopon 12555 TopSpctps 12575 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-i2m1 7849 ax-0lt1 7850 ax-0id 7852 ax-rnegex 7853 ax-pre-ltirr 7856 ax-pre-lttrn 7858 ax-pre-ltadd 7860 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-pnf 7926 df-mnf 7927 df-ltxr 7929 df-inn 8849 df-2 8907 df-3 8908 df-4 8909 df-5 8910 df-6 8911 df-7 8912 df-8 8913 df-9 8914 df-ndx 12340 df-slot 12341 df-base 12343 df-tset 12418 df-rest 12500 df-topn 12501 df-top 12543 df-topon 12556 df-topsp 12576 |
This theorem is referenced by: eltpsi 12586 stoig 12720 |
Copyright terms: Public domain | W3C validator |