| Mathbox for Jeff Hankins |
< Previous
Next >
Nearby theorems |
||
| 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 36339 | . . 3 ⊢ Rel Fne | |
| 2 | 1 | brrelex2i 5742 | . 2 ⊢ (𝐴Fne𝐵 → 𝐵 ∈ V) |
| 3 | simpl 482 | . . . . 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 2800 | . . . 4 ⊢ ((𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)) → ∪ 𝐴 = ∪ 𝐵) |
| 7 | fvex 6919 | . . . . . . 7 ⊢ (topGen‘𝐵) ∈ V | |
| 8 | 7 | ssex 5321 | . . . . . 6 ⊢ (𝐴 ⊆ (topGen‘𝐵) → 𝐴 ∈ V) |
| 9 | 8 | adantl 481 | . . . . 5 ⊢ ((𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)) → 𝐴 ∈ V) |
| 10 | 9 | uniexd 7762 | . . . 4 ⊢ ((𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)) → ∪ 𝐴 ∈ V) |
| 11 | 6, 10 | eqeltrrd 2842 | . . 3 ⊢ ((𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)) → ∪ 𝐵 ∈ V) |
| 12 | uniexb 7784 | . . 3 ⊢ (𝐵 ∈ V ↔ ∪ 𝐵 ∈ V) | |
| 13 | 11, 12 | sylibr 234 | . 2 ⊢ ((𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)) → 𝐵 ∈ V) |
| 14 | 4, 5 | isfne 36340 | . . 3 ⊢ (𝐵 ∈ V → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝐴 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)))) |
| 15 | dfss3 3972 | . . . . 5 ⊢ (𝐴 ⊆ (topGen‘𝐵) ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ (topGen‘𝐵)) | |
| 16 | eltg 22964 | . . . . . 6 ⊢ (𝐵 ∈ V → (𝑥 ∈ (topGen‘𝐵) ↔ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥))) | |
| 17 | 16 | ralbidv 3178 | . . . . 5 ⊢ (𝐵 ∈ V → (∀𝑥 ∈ 𝐴 𝑥 ∈ (topGen‘𝐵) ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥))) |
| 18 | 15, 17 | bitrid 283 | . . . 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 282 | . 2 ⊢ (𝐵 ∈ V → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵)))) |
| 21 | 2, 13, 20 | pm5.21nii 378 | 1 ⊢ (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ 𝐴 ⊆ (topGen‘𝐵))) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 ∩ cin 3950 ⊆ wss 3951 𝒫 cpw 4600 ∪ cuni 4907 class class class wbr 5143 ‘cfv 6561 topGenctg 17482 Fnecfne 36337 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-topgen 17488 df-fne 36338 |
| This theorem is referenced by: isfne4b 36342 isfne2 36343 isfne3 36344 fnebas 36345 fnetg 36346 topfne 36355 fnemeet1 36367 fnemeet2 36368 fnejoin1 36369 fnejoin2 36370 |
| Copyright terms: Public domain | W3C validator |