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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 12192 | . . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐴 ∈ 𝐽) | |
2 | eltpsi.k | . . . . . 6 ⊢ 𝐾 = {〈(Base‘ndx), 𝐴〉, 〈(TopSet‘ndx), 𝐽〉} | |
3 | df-tset 12040 | . . . . . 6 ⊢ TopSet = Slot 9 | |
4 | 1lt9 8924 | . . . . . 6 ⊢ 1 < 9 | |
5 | 9nn 8888 | . . . . . 6 ⊢ 9 ∈ ℕ | |
6 | 2, 3, 4, 5 | 2stropg 12061 | . . . . 5 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐽 ∈ (TopOn‘𝐴)) → 𝐽 = (TopSet‘𝐾)) |
7 | 1, 6 | mpancom 418 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐽 = (TopSet‘𝐾)) |
8 | 2, 3, 4, 5 | 2strbasg 12060 | . . . . . 6 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐽 ∈ (TopOn‘𝐴)) → 𝐴 = (Base‘𝐾)) |
9 | 1, 8 | mpancom 418 | . . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐴 = (Base‘𝐾)) |
10 | 9 | fveq2d 5425 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝐴) → (TopOn‘𝐴) = (TopOn‘(Base‘𝐾))) |
11 | 7, 10 | eleq12d 2210 | . . 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 2139 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
14 | eqid 2139 | . . 3 ⊢ (TopSet‘𝐾) = (TopSet‘𝐾) | |
15 | 13, 14 | tsettps 12205 | . 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 1331 ∈ wcel 1480 {cpr 3528 〈cop 3530 ‘cfv 5123 9c9 8778 ndxcnx 11956 Basecbs 11959 TopSetcts 12027 TopOnctopon 12177 TopSpctps 12197 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-i2m1 7725 ax-0lt1 7726 ax-0id 7728 ax-rnegex 7729 ax-pre-ltirr 7732 ax-pre-lttrn 7734 ax-pre-ltadd 7736 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-pnf 7802 df-mnf 7803 df-ltxr 7805 df-inn 8721 df-2 8779 df-3 8780 df-4 8781 df-5 8782 df-6 8783 df-7 8784 df-8 8785 df-9 8786 df-ndx 11962 df-slot 11963 df-base 11965 df-tset 12040 df-rest 12122 df-topn 12123 df-top 12165 df-topon 12178 df-topsp 12198 |
This theorem is referenced by: eltpsi 12208 stoig 12342 |
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