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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 5170 | . 2 ⊢ (𝐵 ∈ 𝑉 → 𝒫 𝐵 ∈ V) | |
2 | inss1 4125 | . . . . 5 ⊢ (𝐵 ∩ 𝒫 𝑥) ⊆ 𝐵 | |
3 | vpwex 5169 | . . . . . . 7 ⊢ 𝒫 𝑥 ∈ V | |
4 | 3 | inex2 5113 | . . . . . 6 ⊢ (𝐵 ∩ 𝒫 𝑥) ∈ V |
5 | 4 | elpw 4459 | . . . . 5 ⊢ ((𝐵 ∩ 𝒫 𝑥) ∈ 𝒫 𝐵 ↔ (𝐵 ∩ 𝒫 𝑥) ⊆ 𝐵) |
6 | 2, 5 | mpbir 232 | . . . 4 ⊢ (𝐵 ∩ 𝒫 𝑥) ∈ 𝒫 𝐵 |
7 | 6 | a1i 11 | . . 3 ⊢ (𝑥 ∈ (topGen‘𝐵) → (𝐵 ∩ 𝒫 𝑥) ∈ 𝒫 𝐵) |
8 | unieq 4753 | . . . . . . 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 21252 | . . . . . . 7 ⊢ (𝑥 ∈ (topGen‘𝐵) → 𝑥 = ∪ (𝐵 ∩ 𝒫 𝑥)) | |
11 | 10 | ad2antrr 722 | . . . . . 6 ⊢ (((𝑥 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ (topGen‘𝐵)) ∧ (𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦)) → 𝑥 = ∪ (𝐵 ∩ 𝒫 𝑥)) |
12 | eltg4i 21252 | . . . . . . 7 ⊢ (𝑦 ∈ (topGen‘𝐵) → 𝑦 = ∪ (𝐵 ∩ 𝒫 𝑦)) | |
13 | 12 | ad2antlr 723 | . . . . . 6 ⊢ (((𝑥 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ (topGen‘𝐵)) ∧ (𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦)) → 𝑦 = ∪ (𝐵 ∩ 𝒫 𝑦)) |
14 | 9, 11, 13 | 3eqtr4d 2841 | . . . . 5 ⊢ (((𝑥 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ (topGen‘𝐵)) ∧ (𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦)) → 𝑥 = 𝑦) |
15 | 14 | ex 413 | . . . 4 ⊢ ((𝑥 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ (topGen‘𝐵)) → ((𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦) → 𝑥 = 𝑦)) |
16 | pweq 4456 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝒫 𝑥 = 𝒫 𝑦) | |
17 | 16 | ineq2d 4109 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦)) |
18 | 15, 17 | impbid1 226 | . . 3 ⊢ ((𝑥 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ (topGen‘𝐵)) → ((𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦) ↔ 𝑥 = 𝑦)) |
19 | 7, 18 | dom2 8400 | . 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 396 = wceq 1522 ∈ wcel 2081 Vcvv 3437 ∩ cin 3858 ⊆ wss 3859 𝒫 cpw 4453 ∪ cuni 4745 class class class wbr 4962 ‘cfv 6225 ≼ cdom 8355 topGenctg 16540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-id 5348 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-dom 8359 df-topgen 16546 |
This theorem is referenced by: 2ndcredom 21742 kelac2lem 39149 |
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