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Mirrors > Home > ILE Home > Th. List > tgdom | GIF version |
Description: A space has no more open sets than subsets of a basis. (Contributed by Stefan O'Rear, 22-Feb-2015.) (Revised by Mario Carneiro, 9-Apr-2015.) |
Ref | Expression |
---|---|
tgdom | ⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) ≼ 𝒫 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwexg 4141 | . 2 ⊢ (𝐵 ∈ 𝑉 → 𝒫 𝐵 ∈ V) | |
2 | inss1 3327 | . . . . 5 ⊢ (𝐵 ∩ 𝒫 𝑥) ⊆ 𝐵 | |
3 | vpwex 4140 | . . . . . . 7 ⊢ 𝒫 𝑥 ∈ V | |
4 | 3 | inex2 4099 | . . . . . 6 ⊢ (𝐵 ∩ 𝒫 𝑥) ∈ V |
5 | 4 | elpw 3549 | . . . . 5 ⊢ ((𝐵 ∩ 𝒫 𝑥) ∈ 𝒫 𝐵 ↔ (𝐵 ∩ 𝒫 𝑥) ⊆ 𝐵) |
6 | 2, 5 | mpbir 145 | . . . 4 ⊢ (𝐵 ∩ 𝒫 𝑥) ∈ 𝒫 𝐵 |
7 | 6 | a1i 9 | . . 3 ⊢ (𝑥 ∈ (topGen‘𝐵) → (𝐵 ∩ 𝒫 𝑥) ∈ 𝒫 𝐵) |
8 | unieq 3781 | . . . . . . 7 ⊢ ((𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦) → ∪ (𝐵 ∩ 𝒫 𝑥) = ∪ (𝐵 ∩ 𝒫 𝑦)) | |
9 | 8 | adantl 275 | . . . . . 6 ⊢ (((𝑥 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ (topGen‘𝐵)) ∧ (𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦)) → ∪ (𝐵 ∩ 𝒫 𝑥) = ∪ (𝐵 ∩ 𝒫 𝑦)) |
10 | eltg4i 12442 | . . . . . . 7 ⊢ (𝑥 ∈ (topGen‘𝐵) → 𝑥 = ∪ (𝐵 ∩ 𝒫 𝑥)) | |
11 | 10 | ad2antrr 480 | . . . . . 6 ⊢ (((𝑥 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ (topGen‘𝐵)) ∧ (𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦)) → 𝑥 = ∪ (𝐵 ∩ 𝒫 𝑥)) |
12 | eltg4i 12442 | . . . . . . 7 ⊢ (𝑦 ∈ (topGen‘𝐵) → 𝑦 = ∪ (𝐵 ∩ 𝒫 𝑦)) | |
13 | 12 | ad2antlr 481 | . . . . . 6 ⊢ (((𝑥 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ (topGen‘𝐵)) ∧ (𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦)) → 𝑦 = ∪ (𝐵 ∩ 𝒫 𝑦)) |
14 | 9, 11, 13 | 3eqtr4d 2200 | . . . . 5 ⊢ (((𝑥 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ (topGen‘𝐵)) ∧ (𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦)) → 𝑥 = 𝑦) |
15 | 14 | ex 114 | . . . 4 ⊢ ((𝑥 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ (topGen‘𝐵)) → ((𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦) → 𝑥 = 𝑦)) |
16 | pweq 3546 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝒫 𝑥 = 𝒫 𝑦) | |
17 | 16 | ineq2d 3308 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦)) |
18 | 15, 17 | impbid1 141 | . . 3 ⊢ ((𝑥 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ (topGen‘𝐵)) → ((𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦) ↔ 𝑥 = 𝑦)) |
19 | 7, 18 | dom2 6717 | . 2 ⊢ (𝒫 𝐵 ∈ V → (topGen‘𝐵) ≼ 𝒫 𝐵) |
20 | 1, 19 | syl 14 | 1 ⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) ≼ 𝒫 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1335 ∈ wcel 2128 Vcvv 2712 ∩ cin 3101 ⊆ wss 3102 𝒫 cpw 3543 ∪ cuni 3772 class class class wbr 3965 ‘cfv 5169 ≼ cdom 6681 topGenctg 12353 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-id 4253 df-xp 4591 df-rel 4592 df-cnv 4593 df-co 4594 df-dm 4595 df-rn 4596 df-res 4597 df-ima 4598 df-iota 5134 df-fun 5171 df-fn 5172 df-f 5173 df-f1 5174 df-fo 5175 df-f1o 5176 df-fv 5177 df-dom 6684 df-topgen 12359 |
This theorem is referenced by: (None) |
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