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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 4099 | . 2 ⊢ (𝐵 ∈ 𝑉 → 𝒫 𝐵 ∈ V) | |
2 | inss1 3291 | . . . . 5 ⊢ (𝐵 ∩ 𝒫 𝑥) ⊆ 𝐵 | |
3 | vpwex 4098 | . . . . . . 7 ⊢ 𝒫 𝑥 ∈ V | |
4 | 3 | inex2 4058 | . . . . . 6 ⊢ (𝐵 ∩ 𝒫 𝑥) ∈ V |
5 | 4 | elpw 3511 | . . . . 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 3740 | . . . . . . 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 12213 | . . . . . . 7 ⊢ (𝑥 ∈ (topGen‘𝐵) → 𝑥 = ∪ (𝐵 ∩ 𝒫 𝑥)) | |
11 | 10 | ad2antrr 479 | . . . . . 6 ⊢ (((𝑥 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ (topGen‘𝐵)) ∧ (𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦)) → 𝑥 = ∪ (𝐵 ∩ 𝒫 𝑥)) |
12 | eltg4i 12213 | . . . . . . 7 ⊢ (𝑦 ∈ (topGen‘𝐵) → 𝑦 = ∪ (𝐵 ∩ 𝒫 𝑦)) | |
13 | 12 | ad2antlr 480 | . . . . . 6 ⊢ (((𝑥 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ (topGen‘𝐵)) ∧ (𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦)) → 𝑦 = ∪ (𝐵 ∩ 𝒫 𝑦)) |
14 | 9, 11, 13 | 3eqtr4d 2180 | . . . . 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 3508 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝒫 𝑥 = 𝒫 𝑦) | |
17 | 16 | ineq2d 3272 | . . . 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 6662 | . 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 1331 ∈ wcel 1480 Vcvv 2681 ∩ cin 3065 ⊆ wss 3066 𝒫 cpw 3505 ∪ cuni 3731 class class class wbr 3924 ‘cfv 5118 ≼ cdom 6626 topGenctg 12124 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-dom 6629 df-topgen 12130 |
This theorem is referenced by: (None) |
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