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Mirrors > Home > MPE Home > Th. List > tgss | Structured version Visualization version GIF version |
Description: Subset relation for generated topologies. (Contributed by NM, 7-May-2007.) |
Ref | Expression |
---|---|
tgss | ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐶) → (topGen‘𝐵) ⊆ (topGen‘𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrin 4210 | . . . . . 6 ⊢ (𝐵 ⊆ 𝐶 → (𝐵 ∩ 𝒫 𝑥) ⊆ (𝐶 ∩ 𝒫 𝑥)) | |
2 | 1 | unissd 4848 | . . . . 5 ⊢ (𝐵 ⊆ 𝐶 → ∪ (𝐵 ∩ 𝒫 𝑥) ⊆ ∪ (𝐶 ∩ 𝒫 𝑥)) |
3 | sstr2 3974 | . . . . 5 ⊢ (𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥) → (∪ (𝐵 ∩ 𝒫 𝑥) ⊆ ∪ (𝐶 ∩ 𝒫 𝑥) → 𝑥 ⊆ ∪ (𝐶 ∩ 𝒫 𝑥))) | |
4 | 2, 3 | syl5com 31 | . . . 4 ⊢ (𝐵 ⊆ 𝐶 → (𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥) → 𝑥 ⊆ ∪ (𝐶 ∩ 𝒫 𝑥))) |
5 | 4 | adantl 484 | . . 3 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐶) → (𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥) → 𝑥 ⊆ ∪ (𝐶 ∩ 𝒫 𝑥))) |
6 | ssexg 5227 | . . . . 5 ⊢ ((𝐵 ⊆ 𝐶 ∧ 𝐶 ∈ 𝑉) → 𝐵 ∈ V) | |
7 | 6 | ancoms 461 | . . . 4 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐶) → 𝐵 ∈ V) |
8 | eltg 21565 | . . . 4 ⊢ (𝐵 ∈ V → (𝑥 ∈ (topGen‘𝐵) ↔ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥))) | |
9 | 7, 8 | syl 17 | . . 3 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐶) → (𝑥 ∈ (topGen‘𝐵) ↔ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥))) |
10 | eltg 21565 | . . . 4 ⊢ (𝐶 ∈ 𝑉 → (𝑥 ∈ (topGen‘𝐶) ↔ 𝑥 ⊆ ∪ (𝐶 ∩ 𝒫 𝑥))) | |
11 | 10 | adantr 483 | . . 3 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐶) → (𝑥 ∈ (topGen‘𝐶) ↔ 𝑥 ⊆ ∪ (𝐶 ∩ 𝒫 𝑥))) |
12 | 5, 9, 11 | 3imtr4d 296 | . 2 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐶) → (𝑥 ∈ (topGen‘𝐵) → 𝑥 ∈ (topGen‘𝐶))) |
13 | 12 | ssrdv 3973 | 1 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐶) → (topGen‘𝐵) ⊆ (topGen‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2114 Vcvv 3494 ∩ cin 3935 ⊆ wss 3936 𝒫 cpw 4539 ∪ cuni 4838 ‘cfv 6355 topGenctg 16711 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-iota 6314 df-fun 6357 df-fv 6363 df-topgen 16717 |
This theorem is referenced by: tgidm 21588 tgss3 21594 basgen 21596 2basgen 21598 tgfiss 21599 bastop1 21601 lecldbas 21827 txss12 22213 xrtgioo 23414 |
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