Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > bastop | Structured version Visualization version GIF version |
Description: Two ways to express that a basis is a topology. (Contributed by NM, 18-Jul-2006.) |
Ref | Expression |
---|---|
bastop | ⊢ (𝐵 ∈ TopBases → (𝐵 ∈ Top ↔ (topGen‘𝐵) = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgtop 21584 | . 2 ⊢ (𝐵 ∈ Top → (topGen‘𝐵) = 𝐵) | |
2 | tgcl 21580 | . . 3 ⊢ (𝐵 ∈ TopBases → (topGen‘𝐵) ∈ Top) | |
3 | eleq1 2903 | . . 3 ⊢ ((topGen‘𝐵) = 𝐵 → ((topGen‘𝐵) ∈ Top ↔ 𝐵 ∈ Top)) | |
4 | 2, 3 | syl5ibcom 247 | . 2 ⊢ (𝐵 ∈ TopBases → ((topGen‘𝐵) = 𝐵 → 𝐵 ∈ Top)) |
5 | 1, 4 | impbid2 228 | 1 ⊢ (𝐵 ∈ TopBases → (𝐵 ∈ Top ↔ (topGen‘𝐵) = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1536 ∈ wcel 2113 ‘cfv 6358 topGenctg 16714 Topctop 21504 TopBasesctb 21556 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-iota 6317 df-fun 6360 df-fv 6366 df-topgen 16720 df-top 21505 df-bases 21557 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |