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Mirrors > Home > MPE Home > Th. List > Mathboxes > isfne4b | Structured version Visualization version GIF version |
Description: A condition for a topology to be finer than another. (Contributed by Jeff Hankins, 28-Sep-2009.) (Revised by Mario Carneiro, 11-Sep-2015.) |
Ref | Expression |
---|---|
isfne.1 | ⊢ 𝑋 = ∪ 𝐴 |
isfne.2 | ⊢ 𝑌 = ∪ 𝐵 |
Ref | Expression |
---|---|
isfne4b | ⊢ (𝐵 ∈ 𝑉 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 487 | . . . . . . 7 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑋 = 𝑌) → 𝑋 = 𝑌) | |
2 | isfne.1 | . . . . . . 7 ⊢ 𝑋 = ∪ 𝐴 | |
3 | isfne.2 | . . . . . . 7 ⊢ 𝑌 = ∪ 𝐵 | |
4 | 1, 2, 3 | 3eqtr3g 2879 | . . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑋 = 𝑌) → ∪ 𝐴 = ∪ 𝐵) |
5 | uniexg 7465 | . . . . . . 7 ⊢ (𝐵 ∈ 𝑉 → ∪ 𝐵 ∈ V) | |
6 | 5 | adantr 483 | . . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑋 = 𝑌) → ∪ 𝐵 ∈ V) |
7 | 4, 6 | eqeltrd 2913 | . . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑋 = 𝑌) → ∪ 𝐴 ∈ V) |
8 | uniexb 7485 | . . . . 5 ⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V) | |
9 | 7, 8 | sylibr 236 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑋 = 𝑌) → 𝐴 ∈ V) |
10 | simpl 485 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑋 = 𝑌) → 𝐵 ∈ 𝑉) | |
11 | tgss3 21593 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) → ((topGen‘𝐴) ⊆ (topGen‘𝐵) ↔ 𝐴 ⊆ (topGen‘𝐵))) | |
12 | 9, 10, 11 | syl2anc 586 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑋 = 𝑌) → ((topGen‘𝐴) ⊆ (topGen‘𝐵) ↔ 𝐴 ⊆ (topGen‘𝐵))) |
13 | 12 | pm5.32da 581 | . 2 ⊢ (𝐵 ∈ 𝑉 → ((𝑋 = 𝑌 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵)) ↔ (𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)))) |
14 | 2, 3 | isfne4 33688 | . 2 ⊢ (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵))) |
15 | 13, 14 | syl6rbbr 292 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐵)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ⊆ wss 3935 ∪ cuni 4837 class class class wbr 5065 ‘cfv 6354 topGenctg 16710 Fnecfne 33684 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-iota 6313 df-fun 6356 df-fv 6362 df-topgen 16716 df-fne 33685 |
This theorem is referenced by: fnetr 33699 fneval 33700 |
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