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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 31503 | . . 3 ⊢ Rel Fne | |
2 | 1 | brrelex2i 5083 | . 2 ⊢ (𝐴Fne𝐵 → 𝐵 ∈ V) |
3 | simpl 472 | . . . . 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 2667 | . . . 4 ⊢ ((𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)) → ∪ 𝐴 = ∪ 𝐵) |
7 | fvex 6113 | . . . . . . 7 ⊢ (topGen‘𝐵) ∈ V | |
8 | 7 | ssex 4730 | . . . . . 6 ⊢ (𝐴 ⊆ (topGen‘𝐵) → 𝐴 ∈ V) |
9 | 8 | adantl 481 | . . . . 5 ⊢ ((𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)) → 𝐴 ∈ V) |
10 | uniexb 6866 | . . . . 5 ⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V) | |
11 | 9, 10 | sylib 207 | . . . 4 ⊢ ((𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)) → ∪ 𝐴 ∈ V) |
12 | 6, 11 | eqeltrrd 2689 | . . 3 ⊢ ((𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)) → ∪ 𝐵 ∈ V) |
13 | uniexb 6866 | . . 3 ⊢ (𝐵 ∈ V ↔ ∪ 𝐵 ∈ V) | |
14 | 12, 13 | sylibr 223 | . 2 ⊢ ((𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)) → 𝐵 ∈ V) |
15 | 4, 5 | isfne 31504 | . . 3 ⊢ (𝐵 ∈ V → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝐴 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)))) |
16 | dfss3 3558 | . . . . 5 ⊢ (𝐴 ⊆ (topGen‘𝐵) ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ (topGen‘𝐵)) | |
17 | eltg 20572 | . . . . . 6 ⊢ (𝐵 ∈ V → (𝑥 ∈ (topGen‘𝐵) ↔ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥))) | |
18 | 17 | ralbidv 2969 | . . . . 5 ⊢ (𝐵 ∈ V → (∀𝑥 ∈ 𝐴 𝑥 ∈ (topGen‘𝐵) ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥))) |
19 | 16, 18 | syl5bb 271 | . . . 4 ⊢ (𝐵 ∈ V → (𝐴 ⊆ (topGen‘𝐵) ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥))) |
20 | 19 | anbi2d 736 | . . 3 ⊢ (𝐵 ∈ V → ((𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)) ↔ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝐴 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)))) |
21 | 15, 20 | bitr4d 270 | . 2 ⊢ (𝐵 ∈ V → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)))) |
22 | 2, 14, 21 | pm5.21nii 367 | 1 ⊢ (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ∩ cin 3539 ⊆ wss 3540 𝒫 cpw 4108 ∪ cuni 4372 class class class wbr 4583 ‘cfv 5804 topGenctg 15921 Fnecfne 31501 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-topgen 15927 df-fne 31502 |
This theorem is referenced by: isfne4b 31506 isfne2 31507 isfne3 31508 fnebas 31509 fnetg 31510 topfne 31519 fnemeet1 31531 fnemeet2 31532 fnejoin1 31533 fnejoin2 31534 |
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