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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 32921 | . . 3 ⊢ Rel Fne | |
2 | 1 | brrelex2i 5407 | . 2 ⊢ (𝐴Fne𝐵 → 𝐵 ∈ V) |
3 | simpl 476 | . . . . 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 2836 | . . . 4 ⊢ ((𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)) → ∪ 𝐴 = ∪ 𝐵) |
7 | fvex 6459 | . . . . . . 7 ⊢ (topGen‘𝐵) ∈ V | |
8 | 7 | ssex 5039 | . . . . . 6 ⊢ (𝐴 ⊆ (topGen‘𝐵) → 𝐴 ∈ V) |
9 | 8 | adantl 475 | . . . . 5 ⊢ ((𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)) → 𝐴 ∈ V) |
10 | uniexb 7250 | . . . . 5 ⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V) | |
11 | 9, 10 | sylib 210 | . . . 4 ⊢ ((𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)) → ∪ 𝐴 ∈ V) |
12 | 6, 11 | eqeltrrd 2859 | . . 3 ⊢ ((𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)) → ∪ 𝐵 ∈ V) |
13 | uniexb 7250 | . . 3 ⊢ (𝐵 ∈ V ↔ ∪ 𝐵 ∈ V) | |
14 | 12, 13 | sylibr 226 | . 2 ⊢ ((𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)) → 𝐵 ∈ V) |
15 | 4, 5 | isfne 32922 | . . 3 ⊢ (𝐵 ∈ V → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝐴 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)))) |
16 | dfss3 3809 | . . . . 5 ⊢ (𝐴 ⊆ (topGen‘𝐵) ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ (topGen‘𝐵)) | |
17 | eltg 21169 | . . . . . 6 ⊢ (𝐵 ∈ V → (𝑥 ∈ (topGen‘𝐵) ↔ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥))) | |
18 | 17 | ralbidv 3167 | . . . . 5 ⊢ (𝐵 ∈ V → (∀𝑥 ∈ 𝐴 𝑥 ∈ (topGen‘𝐵) ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥))) |
19 | 16, 18 | syl5bb 275 | . . . 4 ⊢ (𝐵 ∈ V → (𝐴 ⊆ (topGen‘𝐵) ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥))) |
20 | 19 | anbi2d 622 | . . 3 ⊢ (𝐵 ∈ V → ((𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)) ↔ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝐴 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)))) |
21 | 15, 20 | bitr4d 274 | . 2 ⊢ (𝐵 ∈ V → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)))) |
22 | 2, 14, 21 | pm5.21nii 370 | 1 ⊢ (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2106 ∀wral 3089 Vcvv 3397 ∩ cin 3790 ⊆ wss 3791 𝒫 cpw 4378 ∪ cuni 4671 class class class wbr 4886 ‘cfv 6135 topGenctg 16484 Fnecfne 32919 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-sbc 3652 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-iota 6099 df-fun 6137 df-fv 6143 df-topgen 16490 df-fne 32920 |
This theorem is referenced by: isfne4b 32924 isfne2 32925 isfne3 32926 fnebas 32927 fnetg 32928 topfne 32937 fnemeet1 32949 fnemeet2 32950 fnejoin1 32951 fnejoin2 32952 |
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