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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 36354 | . . . 4 ⊢ ∼ Er V | 
| 3 | errel 8754 | . . . 4 ⊢ ( ∼ Er V → Rel ∼ ) | |
| 4 | 2, 3 | ax-mp 5 | . . 3 ⊢ Rel ∼ | 
| 5 | relelec 8792 | . . 3 ⊢ (Rel ∼ → (𝐴 ∈ [𝐽] ∼ ↔ 𝐽 ∼ 𝐴)) | |
| 6 | 4, 5 | ax-mp 5 | . 2 ⊢ (𝐴 ∈ [𝐽] ∼ ↔ 𝐽 ∼ 𝐴) | 
| 7 | 4 | brrelex2i 5742 | . . . 4 ⊢ (𝐽 ∼ 𝐴 → 𝐴 ∈ V) | 
| 8 | 7 | a1i 11 | . . 3 ⊢ (𝐽 ∈ Top → (𝐽 ∼ 𝐴 → 𝐴 ∈ V)) | 
| 9 | eleq1 2829 | . . . . . . 7 ⊢ ((topGen‘𝐴) = 𝐽 → ((topGen‘𝐴) ∈ Top ↔ 𝐽 ∈ Top)) | |
| 10 | 9 | biimparc 479 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (topGen‘𝐴) = 𝐽) → (topGen‘𝐴) ∈ Top) | 
| 11 | tgclb 22977 | . . . . . 6 ⊢ (𝐴 ∈ TopBases ↔ (topGen‘𝐴) ∈ Top) | |
| 12 | 10, 11 | sylibr 234 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ (topGen‘𝐴) = 𝐽) → 𝐴 ∈ TopBases) | 
| 13 | elex 3501 | . . . . 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 36353 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝐽 ∼ 𝐴 ↔ (topGen‘𝐽) = (topGen‘𝐴))) | 
| 17 | tgtop 22980 | . . . . . . . 8 ⊢ (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽) | |
| 18 | 17 | eqeq1d 2739 | . . . . . . 7 ⊢ (𝐽 ∈ Top → ((topGen‘𝐽) = (topGen‘𝐴) ↔ 𝐽 = (topGen‘𝐴))) | 
| 19 | eqcom 2744 | . . . . . . 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 1540 ∈ wcel 2108 Vcvv 3480 ∩ cin 3950 class class class wbr 5143 ◡ccnv 5684 Rel wrel 5690 ‘cfv 6561 Er wer 8742 [cec 8743 topGenctg 17482 Topctop 22899 TopBasesctb 22952 Fnecfne 36337 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fv 6569 df-er 8745 df-ec 8747 df-topgen 17488 df-top 22900 df-bases 22953 df-fne 36338 | 
| This theorem is referenced by: (None) | 
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