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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 31518 | . . . 4 ⊢ ∼ Er V |
3 | errel 7638 | . . . 4 ⊢ ( ∼ Er V → Rel ∼ ) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ Rel ∼ |
5 | relelec 7674 | . . 3 ⊢ (Rel ∼ → (𝐴 ∈ [𝐽] ∼ ↔ 𝐽 ∼ 𝐴)) | |
6 | 4, 5 | ax-mp 5 | . 2 ⊢ (𝐴 ∈ [𝐽] ∼ ↔ 𝐽 ∼ 𝐴) |
7 | 4 | brrelex2i 5083 | . . . 4 ⊢ (𝐽 ∼ 𝐴 → 𝐴 ∈ V) |
8 | 7 | a1i 11 | . . 3 ⊢ (𝐽 ∈ Top → (𝐽 ∼ 𝐴 → 𝐴 ∈ V)) |
9 | eleq1 2676 | . . . . . . 7 ⊢ ((topGen‘𝐴) = 𝐽 → ((topGen‘𝐴) ∈ Top ↔ 𝐽 ∈ Top)) | |
10 | 9 | biimparc 503 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (topGen‘𝐴) = 𝐽) → (topGen‘𝐴) ∈ Top) |
11 | tgclb 20585 | . . . . . 6 ⊢ (𝐴 ∈ TopBases ↔ (topGen‘𝐴) ∈ Top) | |
12 | 10, 11 | sylibr 223 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ (topGen‘𝐴) = 𝐽) → 𝐴 ∈ TopBases) |
13 | elex 3185 | . . . . 5 ⊢ (𝐴 ∈ TopBases → 𝐴 ∈ V) | |
14 | 12, 13 | syl 17 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ (topGen‘𝐴) = 𝐽) → 𝐴 ∈ V) |
15 | 14 | ex 449 | . . 3 ⊢ (𝐽 ∈ Top → ((topGen‘𝐴) = 𝐽 → 𝐴 ∈ V)) |
16 | 1 | fneval 31517 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝐽 ∼ 𝐴 ↔ (topGen‘𝐽) = (topGen‘𝐴))) |
17 | tgtop 20588 | . . . . . . . 8 ⊢ (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽) | |
18 | 17 | eqeq1d 2612 | . . . . . . 7 ⊢ (𝐽 ∈ Top → ((topGen‘𝐽) = (topGen‘𝐴) ↔ 𝐽 = (topGen‘𝐴))) |
19 | eqcom 2617 | . . . . . . 7 ⊢ (𝐽 = (topGen‘𝐴) ↔ (topGen‘𝐴) = 𝐽) | |
20 | 18, 19 | syl6bb 275 | . . . . . 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 267 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝐽 ∼ 𝐴 ↔ (topGen‘𝐴) = 𝐽)) |
23 | 22 | ex 449 | . . 3 ⊢ (𝐽 ∈ Top → (𝐴 ∈ V → (𝐽 ∼ 𝐴 ↔ (topGen‘𝐴) = 𝐽))) |
24 | 8, 15, 23 | pm5.21ndd 368 | . 2 ⊢ (𝐽 ∈ Top → (𝐽 ∼ 𝐴 ↔ (topGen‘𝐴) = 𝐽)) |
25 | 6, 24 | syl5bb 271 | 1 ⊢ (𝐽 ∈ Top → (𝐴 ∈ [𝐽] ∼ ↔ (topGen‘𝐴) = 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∩ cin 3539 class class class wbr 4583 ◡ccnv 5037 Rel wrel 5043 ‘cfv 5804 Er wer 7626 [cec 7627 topGenctg 15921 Topctop 20517 TopBasesctb 20520 Fnecfne 31501 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fv 5812 df-er 7629 df-ec 7631 df-topgen 15927 df-top 20521 df-bases 20522 df-fne 31502 |
This theorem is referenced by: (None) |
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