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Mirrors > Home > ILE Home > Th. List > tgss3 | GIF version |
Description: A criterion for determining whether one topology is finer than another. Lemma 2.2 of [Munkres] p. 80 using abbreviations. (Contributed by NM, 20-Jul-2006.) (Proof shortened by Mario Carneiro, 2-Sep-2015.) |
Ref | Expression |
---|---|
tgss3 | ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((topGen‘𝐵) ⊆ (topGen‘𝐶) ↔ 𝐵 ⊆ (topGen‘𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bastg 12230 | . . . 4 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ⊆ (topGen‘𝐵)) | |
2 | 1 | adantr 274 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → 𝐵 ⊆ (topGen‘𝐵)) |
3 | sstr2 3104 | . . 3 ⊢ (𝐵 ⊆ (topGen‘𝐵) → ((topGen‘𝐵) ⊆ (topGen‘𝐶) → 𝐵 ⊆ (topGen‘𝐶))) | |
4 | 2, 3 | syl 14 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((topGen‘𝐵) ⊆ (topGen‘𝐶) → 𝐵 ⊆ (topGen‘𝐶))) |
5 | tgvalex 12219 | . . . . . 6 ⊢ (𝐶 ∈ 𝑊 → (topGen‘𝐶) ∈ V) | |
6 | tgss 12232 | . . . . . 6 ⊢ (((topGen‘𝐶) ∈ V ∧ 𝐵 ⊆ (topGen‘𝐶)) → (topGen‘𝐵) ⊆ (topGen‘(topGen‘𝐶))) | |
7 | 5, 6 | sylan 281 | . . . . 5 ⊢ ((𝐶 ∈ 𝑊 ∧ 𝐵 ⊆ (topGen‘𝐶)) → (topGen‘𝐵) ⊆ (topGen‘(topGen‘𝐶))) |
8 | 7 | ex 114 | . . . 4 ⊢ (𝐶 ∈ 𝑊 → (𝐵 ⊆ (topGen‘𝐶) → (topGen‘𝐵) ⊆ (topGen‘(topGen‘𝐶)))) |
9 | 8 | adantl 275 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ⊆ (topGen‘𝐶) → (topGen‘𝐵) ⊆ (topGen‘(topGen‘𝐶)))) |
10 | tgidm 12243 | . . . . 5 ⊢ (𝐶 ∈ 𝑊 → (topGen‘(topGen‘𝐶)) = (topGen‘𝐶)) | |
11 | 10 | adantl 275 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (topGen‘(topGen‘𝐶)) = (topGen‘𝐶)) |
12 | 11 | sseq2d 3127 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((topGen‘𝐵) ⊆ (topGen‘(topGen‘𝐶)) ↔ (topGen‘𝐵) ⊆ (topGen‘𝐶))) |
13 | 9, 12 | sylibd 148 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ⊆ (topGen‘𝐶) → (topGen‘𝐵) ⊆ (topGen‘𝐶))) |
14 | 4, 13 | impbid 128 | 1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((topGen‘𝐵) ⊆ (topGen‘𝐶) ↔ 𝐵 ⊆ (topGen‘𝐶))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1331 ∈ wcel 1480 Vcvv 2686 ⊆ wss 3071 ‘cfv 5123 topGenctg 12135 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-topgen 12141 |
This theorem is referenced by: tgss2 12248 2basgeng 12251 xmettxlem 12678 xmettx 12679 |
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