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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 36355 | . . . 4 ⊢ ∼ Er V | 
| 3 | errel 8755 | . . . 4 ⊢ ( ∼ Er V → Rel ∼ ) | |
| 4 | 2, 3 | ax-mp 5 | . . 3 ⊢ Rel ∼ | 
| 5 | relelec 8793 | . . 3 ⊢ (Rel ∼ → (𝐴 ∈ [𝐽] ∼ ↔ 𝐽 ∼ 𝐴)) | |
| 6 | 4, 5 | ax-mp 5 | . 2 ⊢ (𝐴 ∈ [𝐽] ∼ ↔ 𝐽 ∼ 𝐴) | 
| 7 | 4 | brrelex2i 5741 | . . . 4 ⊢ (𝐽 ∼ 𝐴 → 𝐴 ∈ V) | 
| 8 | 7 | a1i 11 | . . 3 ⊢ (𝐽 ∈ Top → (𝐽 ∼ 𝐴 → 𝐴 ∈ V)) | 
| 9 | eleq1 2828 | . . . . . . 7 ⊢ ((topGen‘𝐴) = 𝐽 → ((topGen‘𝐴) ∈ Top ↔ 𝐽 ∈ Top)) | |
| 10 | 9 | biimparc 479 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (topGen‘𝐴) = 𝐽) → (topGen‘𝐴) ∈ Top) | 
| 11 | tgclb 22978 | . . . . . 6 ⊢ (𝐴 ∈ TopBases ↔ (topGen‘𝐴) ∈ Top) | |
| 12 | 10, 11 | sylibr 234 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ (topGen‘𝐴) = 𝐽) → 𝐴 ∈ TopBases) | 
| 13 | elex 3500 | . . . . 5 ⊢ (𝐴 ∈ TopBases → 𝐴 ∈ V) | |
| 14 | 12, 13 | syl 17 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ (topGen‘𝐴) = 𝐽) → 𝐴 ∈ V) | 
| 15 | 14 | ex 412 | . . 3 ⊢ (𝐽 ∈ Top → ((topGen‘𝐴) = 𝐽 → 𝐴 ∈ V)) | 
| 16 | 1 | fneval 36354 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝐽 ∼ 𝐴 ↔ (topGen‘𝐽) = (topGen‘𝐴))) | 
| 17 | tgtop 22981 | . . . . . . . 8 ⊢ (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽) | |
| 18 | 17 | eqeq1d 2738 | . . . . . . 7 ⊢ (𝐽 ∈ Top → ((topGen‘𝐽) = (topGen‘𝐴) ↔ 𝐽 = (topGen‘𝐴))) | 
| 19 | eqcom 2743 | . . . . . . 7 ⊢ (𝐽 = (topGen‘𝐴) ↔ (topGen‘𝐴) = 𝐽) | |
| 20 | 18, 19 | bitrdi 287 | . . . . . 6 ⊢ (𝐽 ∈ Top → ((topGen‘𝐽) = (topGen‘𝐴) ↔ (topGen‘𝐴) = 𝐽)) | 
| 21 | 20 | adantr 480 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → ((topGen‘𝐽) = (topGen‘𝐴) ↔ (topGen‘𝐴) = 𝐽)) | 
| 22 | 16, 21 | bitrd 279 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝐽 ∼ 𝐴 ↔ (topGen‘𝐴) = 𝐽)) | 
| 23 | 22 | ex 412 | . . 3 ⊢ (𝐽 ∈ Top → (𝐴 ∈ V → (𝐽 ∼ 𝐴 ↔ (topGen‘𝐴) = 𝐽))) | 
| 24 | 8, 15, 23 | pm5.21ndd 379 | . 2 ⊢ (𝐽 ∈ Top → (𝐽 ∼ 𝐴 ↔ (topGen‘𝐴) = 𝐽)) | 
| 25 | 6, 24 | bitrid 283 | 1 ⊢ (𝐽 ∈ Top → (𝐴 ∈ [𝐽] ∼ ↔ (topGen‘𝐴) = 𝐽)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 Vcvv 3479 ∩ cin 3949 class class class wbr 5142 ◡ccnv 5683 Rel wrel 5689 ‘cfv 6560 Er wer 8743 [cec 8744 topGenctg 17483 Topctop 22900 TopBasesctb 22953 Fnecfne 36338 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fv 6568 df-er 8746 df-ec 8748 df-topgen 17489 df-top 22901 df-bases 22954 df-fne 36339 | 
| This theorem is referenced by: (None) | 
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