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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 34801 | . . 3 ⊢ Rel Fne | |
2 | 1 | brrelex2i 5689 | . 2 ⊢ (𝐴Fne𝐵 → 𝐵 ∈ V) |
3 | simpl 483 | . . . . 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 2799 | . . . 4 ⊢ ((𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)) → ∪ 𝐴 = ∪ 𝐵) |
7 | fvex 6855 | . . . . . . 7 ⊢ (topGen‘𝐵) ∈ V | |
8 | 7 | ssex 5278 | . . . . . 6 ⊢ (𝐴 ⊆ (topGen‘𝐵) → 𝐴 ∈ V) |
9 | 8 | adantl 482 | . . . . 5 ⊢ ((𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)) → 𝐴 ∈ V) |
10 | 9 | uniexd 7678 | . . . 4 ⊢ ((𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)) → ∪ 𝐴 ∈ V) |
11 | 6, 10 | eqeltrrd 2839 | . . 3 ⊢ ((𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)) → ∪ 𝐵 ∈ V) |
12 | uniexb 7697 | . . 3 ⊢ (𝐵 ∈ V ↔ ∪ 𝐵 ∈ V) | |
13 | 11, 12 | sylibr 233 | . 2 ⊢ ((𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)) → 𝐵 ∈ V) |
14 | 4, 5 | isfne 34802 | . . 3 ⊢ (𝐵 ∈ V → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝐴 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)))) |
15 | dfss3 3932 | . . . . 5 ⊢ (𝐴 ⊆ (topGen‘𝐵) ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ (topGen‘𝐵)) | |
16 | eltg 22305 | . . . . . 6 ⊢ (𝐵 ∈ V → (𝑥 ∈ (topGen‘𝐵) ↔ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥))) | |
17 | 16 | ralbidv 3174 | . . . . 5 ⊢ (𝐵 ∈ V → (∀𝑥 ∈ 𝐴 𝑥 ∈ (topGen‘𝐵) ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥))) |
18 | 15, 17 | bitrid 282 | . . . 4 ⊢ (𝐵 ∈ V → (𝐴 ⊆ (topGen‘𝐵) ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥))) |
19 | 18 | anbi2d 629 | . . 3 ⊢ (𝐵 ∈ V → ((𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)) ↔ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝐴 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)))) |
20 | 14, 19 | bitr4d 281 | . 2 ⊢ (𝐵 ∈ V → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)))) |
21 | 2, 13, 20 | pm5.21nii 379 | 1 ⊢ (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3064 Vcvv 3445 ∩ cin 3909 ⊆ wss 3910 𝒫 cpw 4560 ∪ cuni 4865 class class class wbr 5105 ‘cfv 6496 topGenctg 17318 Fnecfne 34799 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-iota 6448 df-fun 6498 df-fv 6504 df-topgen 17324 df-fne 34800 |
This theorem is referenced by: isfne4b 34804 isfne2 34805 isfne3 34806 fnebas 34807 fnetg 34808 topfne 34817 fnemeet1 34829 fnemeet2 34830 fnejoin1 34831 fnejoin2 34832 |
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