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Mirrors > Home > MPE Home > Th. List > Mathboxes > isfne4 | Structured version Visualization version GIF version |
Description: The predicate "𝐵 is finer than 𝐴 " in terms of the topology generation function. (Contributed by Mario Carneiro, 11-Sep-2015.) |
Ref | Expression |
---|---|
isfne.1 | ⊢ 𝑋 = ∪ 𝐴 |
isfne.2 | ⊢ 𝑌 = ∪ 𝐵 |
Ref | Expression |
---|---|
isfne4 | ⊢ (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnerel 33681 | . . 3 ⊢ Rel Fne | |
2 | 1 | brrelex2i 5603 | . 2 ⊢ (𝐴Fne𝐵 → 𝐵 ∈ V) |
3 | simpl 485 | . . . . 5 ⊢ ((𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)) → 𝑋 = 𝑌) | |
4 | isfne.1 | . . . . 5 ⊢ 𝑋 = ∪ 𝐴 | |
5 | isfne.2 | . . . . 5 ⊢ 𝑌 = ∪ 𝐵 | |
6 | 3, 4, 5 | 3eqtr3g 2879 | . . . 4 ⊢ ((𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)) → ∪ 𝐴 = ∪ 𝐵) |
7 | fvex 6677 | . . . . . . 7 ⊢ (topGen‘𝐵) ∈ V | |
8 | 7 | ssex 5217 | . . . . . 6 ⊢ (𝐴 ⊆ (topGen‘𝐵) → 𝐴 ∈ V) |
9 | 8 | adantl 484 | . . . . 5 ⊢ ((𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)) → 𝐴 ∈ V) |
10 | 9 | uniexd 7462 | . . . 4 ⊢ ((𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)) → ∪ 𝐴 ∈ V) |
11 | 6, 10 | eqeltrrd 2914 | . . 3 ⊢ ((𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)) → ∪ 𝐵 ∈ V) |
12 | uniexb 7480 | . . 3 ⊢ (𝐵 ∈ V ↔ ∪ 𝐵 ∈ V) | |
13 | 11, 12 | sylibr 236 | . 2 ⊢ ((𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)) → 𝐵 ∈ V) |
14 | 4, 5 | isfne 33682 | . . 3 ⊢ (𝐵 ∈ V → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝐴 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)))) |
15 | dfss3 3955 | . . . . 5 ⊢ (𝐴 ⊆ (topGen‘𝐵) ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ (topGen‘𝐵)) | |
16 | eltg 21559 | . . . . . 6 ⊢ (𝐵 ∈ V → (𝑥 ∈ (topGen‘𝐵) ↔ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥))) | |
17 | 16 | ralbidv 3197 | . . . . 5 ⊢ (𝐵 ∈ V → (∀𝑥 ∈ 𝐴 𝑥 ∈ (topGen‘𝐵) ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥))) |
18 | 15, 17 | syl5bb 285 | . . . 4 ⊢ (𝐵 ∈ V → (𝐴 ⊆ (topGen‘𝐵) ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥))) |
19 | 18 | anbi2d 630 | . . 3 ⊢ (𝐵 ∈ V → ((𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)) ↔ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝐴 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)))) |
20 | 14, 19 | bitr4d 284 | . 2 ⊢ (𝐵 ∈ V → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)))) |
21 | 2, 13, 20 | pm5.21nii 382 | 1 ⊢ (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 Vcvv 3494 ∩ cin 3934 ⊆ wss 3935 𝒫 cpw 4538 ∪ cuni 4831 class class class wbr 5058 ‘cfv 6349 topGenctg 16705 Fnecfne 33679 |
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 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
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 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-iota 6308 df-fun 6351 df-fv 6357 df-topgen 16711 df-fne 33680 |
This theorem is referenced by: isfne4b 33684 isfne2 33685 isfne3 33686 fnebas 33687 fnetg 33688 topfne 33697 fnemeet1 33709 fnemeet2 33710 fnejoin1 33711 fnejoin2 33712 |
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