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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.) |
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 | df-tset 16586 | . . . . 5 ⊢ TopSet = Slot 9 | |
3 | 1lt9 11846 | . . . . 5 ⊢ 1 < 9 | |
4 | 9nn 11738 | . . . . 5 ⊢ 9 ∈ ℕ | |
5 | 1, 2, 3, 4 | 2strop 16606 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐽 = (TopSet‘𝐾)) |
6 | toponmax 21536 | . . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐴 ∈ 𝐽) | |
7 | 1, 2, 3, 4 | 2strbas 16605 | . . . . . 6 ⊢ (𝐴 ∈ 𝐽 → 𝐴 = (Base‘𝐾)) |
8 | 6, 7 | syl 17 | . . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐴 = (Base‘𝐾)) |
9 | 8 | fveq2d 6676 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝐴) → (TopOn‘𝐴) = (TopOn‘(Base‘𝐾))) |
10 | 5, 9 | eleq12d 2909 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝐴) → (𝐽 ∈ (TopOn‘𝐴) ↔ (TopSet‘𝐾) ∈ (TopOn‘(Base‘𝐾)))) |
11 | 10 | ibi 269 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝐴) → (TopSet‘𝐾) ∈ (TopOn‘(Base‘𝐾))) |
12 | eqid 2823 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
13 | eqid 2823 | . . 3 ⊢ (TopSet‘𝐾) = (TopSet‘𝐾) | |
14 | 12, 13 | tsettps 21551 | . 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 1537 ∈ wcel 2114 {cpr 4571 〈cop 4575 ‘cfv 6357 9c9 11702 ndxcnx 16482 Basecbs 16485 TopSetcts 16573 TopOnctopon 21520 TopSpctps 21542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-tset 16586 df-rest 16698 df-topn 16699 df-top 21504 df-topon 21521 df-topsp 21543 |
This theorem is referenced by: eltpsi 21554 stoig 21773 |
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