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Mirrors > Home > MPE Home > Th. List > tgdom | Structured version Visualization version 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 4850 | . 2 ⊢ (𝐵 ∈ 𝑉 → 𝒫 𝐵 ∈ V) | |
2 | inss1 3833 | . . . . 5 ⊢ (𝐵 ∩ 𝒫 𝑥) ⊆ 𝐵 | |
3 | vpwex 4849 | . . . . . . 7 ⊢ 𝒫 𝑥 ∈ V | |
4 | 3 | inex2 4800 | . . . . . 6 ⊢ (𝐵 ∩ 𝒫 𝑥) ∈ V |
5 | 4 | elpw 4164 | . . . . 5 ⊢ ((𝐵 ∩ 𝒫 𝑥) ∈ 𝒫 𝐵 ↔ (𝐵 ∩ 𝒫 𝑥) ⊆ 𝐵) |
6 | 2, 5 | mpbir 221 | . . . 4 ⊢ (𝐵 ∩ 𝒫 𝑥) ∈ 𝒫 𝐵 |
7 | 6 | a1i 11 | . . 3 ⊢ (𝑥 ∈ (topGen‘𝐵) → (𝐵 ∩ 𝒫 𝑥) ∈ 𝒫 𝐵) |
8 | unieq 4444 | . . . . . . 7 ⊢ ((𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦) → ∪ (𝐵 ∩ 𝒫 𝑥) = ∪ (𝐵 ∩ 𝒫 𝑦)) | |
9 | 8 | adantl 482 | . . . . . 6 ⊢ (((𝑥 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ (topGen‘𝐵)) ∧ (𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦)) → ∪ (𝐵 ∩ 𝒫 𝑥) = ∪ (𝐵 ∩ 𝒫 𝑦)) |
10 | eltg4i 20764 | . . . . . . 7 ⊢ (𝑥 ∈ (topGen‘𝐵) → 𝑥 = ∪ (𝐵 ∩ 𝒫 𝑥)) | |
11 | 10 | ad2antrr 762 | . . . . . 6 ⊢ (((𝑥 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ (topGen‘𝐵)) ∧ (𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦)) → 𝑥 = ∪ (𝐵 ∩ 𝒫 𝑥)) |
12 | eltg4i 20764 | . . . . . . 7 ⊢ (𝑦 ∈ (topGen‘𝐵) → 𝑦 = ∪ (𝐵 ∩ 𝒫 𝑦)) | |
13 | 12 | ad2antlr 763 | . . . . . 6 ⊢ (((𝑥 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ (topGen‘𝐵)) ∧ (𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦)) → 𝑦 = ∪ (𝐵 ∩ 𝒫 𝑦)) |
14 | 9, 11, 13 | 3eqtr4d 2666 | . . . . 5 ⊢ (((𝑥 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ (topGen‘𝐵)) ∧ (𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦)) → 𝑥 = 𝑦) |
15 | 14 | ex 450 | . . . 4 ⊢ ((𝑥 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ (topGen‘𝐵)) → ((𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦) → 𝑥 = 𝑦)) |
16 | pweq 4161 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝒫 𝑥 = 𝒫 𝑦) | |
17 | 16 | ineq2d 3814 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦)) |
18 | 15, 17 | impbid1 215 | . . 3 ⊢ ((𝑥 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ (topGen‘𝐵)) → ((𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦) ↔ 𝑥 = 𝑦)) |
19 | 7, 18 | dom2 7998 | . 2 ⊢ (𝒫 𝐵 ∈ V → (topGen‘𝐵) ≼ 𝒫 𝐵) |
20 | 1, 19 | syl 17 | 1 ⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) ≼ 𝒫 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∩ cin 3573 ⊆ wss 3574 𝒫 cpw 4158 ∪ cuni 4436 class class class wbr 4653 ‘cfv 5888 ≼ cdom 7953 topGenctg 16098 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-dom 7957 df-topgen 16104 |
This theorem is referenced by: 2ndcredom 21253 kelac2lem 37634 |
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