![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > eltg3i | Structured version Visualization version GIF version |
Description: The union of a set of basic open sets is in the generated topology. (Contributed by Mario Carneiro, 30-Aug-2015.) |
Ref | Expression |
---|---|
eltg3i | ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → ∪ 𝐴 ∈ (topGen‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 485 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → 𝐴 ⊆ 𝐵) | |
2 | pwuni 4781 | . . . 4 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴 | |
3 | ssin 4127 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝒫 ∪ 𝐴) ↔ 𝐴 ⊆ (𝐵 ∩ 𝒫 ∪ 𝐴)) | |
4 | 1, 2, 3 | sylanblc 589 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → 𝐴 ⊆ (𝐵 ∩ 𝒫 ∪ 𝐴)) |
5 | 4 | unissd 4769 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → ∪ 𝐴 ⊆ ∪ (𝐵 ∩ 𝒫 ∪ 𝐴)) |
6 | eltg 21249 | . . 3 ⊢ (𝐵 ∈ 𝑉 → (∪ 𝐴 ∈ (topGen‘𝐵) ↔ ∪ 𝐴 ⊆ ∪ (𝐵 ∩ 𝒫 ∪ 𝐴))) | |
7 | 6 | adantr 481 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → (∪ 𝐴 ∈ (topGen‘𝐵) ↔ ∪ 𝐴 ⊆ ∪ (𝐵 ∩ 𝒫 ∪ 𝐴))) |
8 | 5, 7 | mpbird 258 | 1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → ∪ 𝐴 ∈ (topGen‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∈ wcel 2081 ∩ cin 3858 ⊆ wss 3859 𝒫 cpw 4453 ∪ cuni 4745 ‘cfv 6225 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-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-ral 3110 df-rex 3111 df-rab 3114 df-v 3439 df-sbc 3707 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-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-iota 6189 df-fun 6227 df-fv 6233 df-topgen 16546 |
This theorem is referenced by: eltg3 21254 tgiun 21271 tgidm 21272 tgrest 21451 leordtval2 21504 fnemeet1 33323 fnejoin2 33326 ontgval 33388 |
Copyright terms: Public domain | W3C validator |