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Mirrors > Home > MPE Home > Th. List > bastop1 | Structured version Visualization version GIF version |
Description: A subset of a topology is a basis for the topology iff every member of the topology is a union of members of the basis. We use the idiom "(topGen‘𝐵) = 𝐽 " to express "𝐵 is a basis for topology 𝐽 " since we do not have a separate notation for this. Definition 15.35 of [Schechter] p. 428. (Contributed by NM, 2-Feb-2008.) (Proof shortened by Mario Carneiro, 2-Sep-2015.) |
Ref | Expression |
---|---|
bastop1 | ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝐽) → ((topGen‘𝐵) = 𝐽 ↔ ∀𝑥 ∈ 𝐽 ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgss 21559 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝐽) → (topGen‘𝐵) ⊆ (topGen‘𝐽)) | |
2 | tgtop 21564 | . . . . . 6 ⊢ (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽) | |
3 | 2 | adantr 483 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝐽) → (topGen‘𝐽) = 𝐽) |
4 | 1, 3 | sseqtrd 3995 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝐽) → (topGen‘𝐵) ⊆ 𝐽) |
5 | eqss 3970 | . . . . 5 ⊢ ((topGen‘𝐵) = 𝐽 ↔ ((topGen‘𝐵) ⊆ 𝐽 ∧ 𝐽 ⊆ (topGen‘𝐵))) | |
6 | 5 | baib 538 | . . . 4 ⊢ ((topGen‘𝐵) ⊆ 𝐽 → ((topGen‘𝐵) = 𝐽 ↔ 𝐽 ⊆ (topGen‘𝐵))) |
7 | 4, 6 | syl 17 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝐽) → ((topGen‘𝐵) = 𝐽 ↔ 𝐽 ⊆ (topGen‘𝐵))) |
8 | dfss3 3944 | . . 3 ⊢ (𝐽 ⊆ (topGen‘𝐵) ↔ ∀𝑥 ∈ 𝐽 𝑥 ∈ (topGen‘𝐵)) | |
9 | 7, 8 | syl6bb 289 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝐽) → ((topGen‘𝐵) = 𝐽 ↔ ∀𝑥 ∈ 𝐽 𝑥 ∈ (topGen‘𝐵))) |
10 | ssexg 5213 | . . . . 5 ⊢ ((𝐵 ⊆ 𝐽 ∧ 𝐽 ∈ Top) → 𝐵 ∈ V) | |
11 | 10 | ancoms 461 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝐽) → 𝐵 ∈ V) |
12 | eltg3 21553 | . . . 4 ⊢ (𝐵 ∈ V → (𝑥 ∈ (topGen‘𝐵) ↔ ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦))) | |
13 | 11, 12 | syl 17 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝐽) → (𝑥 ∈ (topGen‘𝐵) ↔ ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦))) |
14 | 13 | ralbidv 3197 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝐽) → (∀𝑥 ∈ 𝐽 𝑥 ∈ (topGen‘𝐵) ↔ ∀𝑥 ∈ 𝐽 ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦))) |
15 | 9, 14 | bitrd 281 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝐽) → ((topGen‘𝐵) = 𝐽 ↔ ∀𝑥 ∈ 𝐽 ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ∀wral 3138 Vcvv 3486 ⊆ wss 3924 ∪ cuni 4824 ‘cfv 6341 topGenctg 16694 Topctop 21484 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3488 df-sbc 3764 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-op 4560 df-uni 4825 df-br 5053 df-opab 5115 df-mpt 5133 df-id 5446 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-iota 6300 df-fun 6343 df-fv 6349 df-topgen 16700 df-top 21485 |
This theorem is referenced by: bastop2 21585 |
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