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Mirrors > Home > MPE Home > Th. List > Mathboxes > topfneec | Structured version Visualization version GIF version |
Description: A cover is equivalent to a topology iff it is a base for that topology. (Contributed by Jeff Hankins, 8-Oct-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.) |
Ref | Expression |
---|---|
topfneec.1 | ⊢ ∼ = (Fne ∩ ◡Fne) |
Ref | Expression |
---|---|
topfneec | ⊢ (𝐽 ∈ Top → (𝐴 ∈ [𝐽] ∼ ↔ (topGen‘𝐴) = 𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | topfneec.1 | . . . . 5 ⊢ ∼ = (Fne ∩ ◡Fne) | |
2 | 1 | fneer 33814 | . . . 4 ⊢ ∼ Er V |
3 | errel 8281 | . . . 4 ⊢ ( ∼ Er V → Rel ∼ ) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ Rel ∼ |
5 | relelec 8317 | . . 3 ⊢ (Rel ∼ → (𝐴 ∈ [𝐽] ∼ ↔ 𝐽 ∼ 𝐴)) | |
6 | 4, 5 | ax-mp 5 | . 2 ⊢ (𝐴 ∈ [𝐽] ∼ ↔ 𝐽 ∼ 𝐴) |
7 | 4 | brrelex2i 5573 | . . . 4 ⊢ (𝐽 ∼ 𝐴 → 𝐴 ∈ V) |
8 | 7 | a1i 11 | . . 3 ⊢ (𝐽 ∈ Top → (𝐽 ∼ 𝐴 → 𝐴 ∈ V)) |
9 | eleq1 2877 | . . . . . . 7 ⊢ ((topGen‘𝐴) = 𝐽 → ((topGen‘𝐴) ∈ Top ↔ 𝐽 ∈ Top)) | |
10 | 9 | biimparc 483 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (topGen‘𝐴) = 𝐽) → (topGen‘𝐴) ∈ Top) |
11 | tgclb 21575 | . . . . . 6 ⊢ (𝐴 ∈ TopBases ↔ (topGen‘𝐴) ∈ Top) | |
12 | 10, 11 | sylibr 237 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ (topGen‘𝐴) = 𝐽) → 𝐴 ∈ TopBases) |
13 | elex 3459 | . . . . 5 ⊢ (𝐴 ∈ TopBases → 𝐴 ∈ V) | |
14 | 12, 13 | syl 17 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ (topGen‘𝐴) = 𝐽) → 𝐴 ∈ V) |
15 | 14 | ex 416 | . . 3 ⊢ (𝐽 ∈ Top → ((topGen‘𝐴) = 𝐽 → 𝐴 ∈ V)) |
16 | 1 | fneval 33813 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝐽 ∼ 𝐴 ↔ (topGen‘𝐽) = (topGen‘𝐴))) |
17 | tgtop 21578 | . . . . . . . 8 ⊢ (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽) | |
18 | 17 | eqeq1d 2800 | . . . . . . 7 ⊢ (𝐽 ∈ Top → ((topGen‘𝐽) = (topGen‘𝐴) ↔ 𝐽 = (topGen‘𝐴))) |
19 | eqcom 2805 | . . . . . . 7 ⊢ (𝐽 = (topGen‘𝐴) ↔ (topGen‘𝐴) = 𝐽) | |
20 | 18, 19 | syl6bb 290 | . . . . . 6 ⊢ (𝐽 ∈ Top → ((topGen‘𝐽) = (topGen‘𝐴) ↔ (topGen‘𝐴) = 𝐽)) |
21 | 20 | adantr 484 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → ((topGen‘𝐽) = (topGen‘𝐴) ↔ (topGen‘𝐴) = 𝐽)) |
22 | 16, 21 | bitrd 282 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝐽 ∼ 𝐴 ↔ (topGen‘𝐴) = 𝐽)) |
23 | 22 | ex 416 | . . 3 ⊢ (𝐽 ∈ Top → (𝐴 ∈ V → (𝐽 ∼ 𝐴 ↔ (topGen‘𝐴) = 𝐽))) |
24 | 8, 15, 23 | pm5.21ndd 384 | . 2 ⊢ (𝐽 ∈ Top → (𝐽 ∼ 𝐴 ↔ (topGen‘𝐴) = 𝐽)) |
25 | 6, 24 | syl5bb 286 | 1 ⊢ (𝐽 ∈ Top → (𝐴 ∈ [𝐽] ∼ ↔ (topGen‘𝐴) = 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∩ cin 3880 class class class wbr 5030 ◡ccnv 5518 Rel wrel 5524 ‘cfv 6324 Er wer 8269 [cec 8270 topGenctg 16703 Topctop 21498 TopBasesctb 21550 Fnecfne 33797 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fv 6332 df-er 8272 df-ec 8274 df-topgen 16709 df-top 21499 df-bases 21551 df-fne 33798 |
This theorem is referenced by: (None) |
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