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Mirrors > Home > MPE Home > Th. List > tgss3 | Structured version Visualization version 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 22332 | . . . 4 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ⊆ (topGen‘𝐵)) | |
2 | 1 | adantr 482 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → 𝐵 ⊆ (topGen‘𝐵)) |
3 | sstr2 3952 | . . 3 ⊢ (𝐵 ⊆ (topGen‘𝐵) → ((topGen‘𝐵) ⊆ (topGen‘𝐶) → 𝐵 ⊆ (topGen‘𝐶))) | |
4 | 2, 3 | syl 17 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((topGen‘𝐵) ⊆ (topGen‘𝐶) → 𝐵 ⊆ (topGen‘𝐶))) |
5 | fvex 6856 | . . . 4 ⊢ (topGen‘𝐶) ∈ V | |
6 | tgss 22334 | . . . 4 ⊢ (((topGen‘𝐶) ∈ V ∧ 𝐵 ⊆ (topGen‘𝐶)) → (topGen‘𝐵) ⊆ (topGen‘(topGen‘𝐶))) | |
7 | 5, 6 | mpan 689 | . . 3 ⊢ (𝐵 ⊆ (topGen‘𝐶) → (topGen‘𝐵) ⊆ (topGen‘(topGen‘𝐶))) |
8 | tgidm 22346 | . . . . 5 ⊢ (𝐶 ∈ 𝑊 → (topGen‘(topGen‘𝐶)) = (topGen‘𝐶)) | |
9 | 8 | adantl 483 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (topGen‘(topGen‘𝐶)) = (topGen‘𝐶)) |
10 | 9 | sseq2d 3977 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((topGen‘𝐵) ⊆ (topGen‘(topGen‘𝐶)) ↔ (topGen‘𝐵) ⊆ (topGen‘𝐶))) |
11 | 7, 10 | imbitrid 243 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ⊆ (topGen‘𝐶) → (topGen‘𝐵) ⊆ (topGen‘𝐶))) |
12 | 4, 11 | impbid 211 | 1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((topGen‘𝐵) ⊆ (topGen‘𝐶) ↔ 𝐵 ⊆ (topGen‘𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3444 ⊆ wss 3911 ‘cfv 6497 topGenctg 17324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-iota 6449 df-fun 6499 df-fv 6505 df-topgen 17330 |
This theorem is referenced by: tgss2 22353 2basgen 22356 isfne4b 34859 |
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