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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 20777 | . 2 ⊢ (𝐵 ∈ Top → (topGen‘𝐵) = 𝐵) | |
2 | tgcl 20773 | . . 3 ⊢ (𝐵 ∈ TopBases → (topGen‘𝐵) ∈ Top) | |
3 | eleq1 2689 | . . 3 ⊢ ((topGen‘𝐵) = 𝐵 → ((topGen‘𝐵) ∈ Top ↔ 𝐵 ∈ Top)) | |
4 | 2, 3 | syl5ibcom 235 | . 2 ⊢ (𝐵 ∈ TopBases → ((topGen‘𝐵) = 𝐵 → 𝐵 ∈ Top)) |
5 | 1, 4 | impbid2 216 | 1 ⊢ (𝐵 ∈ TopBases → (𝐵 ∈ Top ↔ (topGen‘𝐵) = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1483 ∈ wcel 1990 ‘cfv 5888 topGenctg 16098 Topctop 20698 TopBasesctb 20749 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-topgen 16104 df-top 20699 df-bases 20750 |
This theorem is referenced by: (None) |
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