Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > sssalgen | Structured version Visualization version GIF version |
Description: A set is a subset of the sigma-algebra it generates. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
Ref | Expression |
---|---|
sssalgen.1 | ⊢ 𝑆 = (SalGen‘𝑋) |
Ref | Expression |
---|---|
sssalgen | ⊢ (𝑋 ∈ 𝑉 → 𝑋 ⊆ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssint 4861 | . . . 4 ⊢ (𝑋 ⊆ ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ↔ ∀𝑡 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)}𝑋 ⊆ 𝑡) | |
2 | unieq 4816 | . . . . . . . . 9 ⊢ (𝑠 = 𝑡 → ∪ 𝑠 = ∪ 𝑡) | |
3 | 2 | eqeq1d 2738 | . . . . . . . 8 ⊢ (𝑠 = 𝑡 → (∪ 𝑠 = ∪ 𝑋 ↔ ∪ 𝑡 = ∪ 𝑋)) |
4 | sseq2 3913 | . . . . . . . 8 ⊢ (𝑠 = 𝑡 → (𝑋 ⊆ 𝑠 ↔ 𝑋 ⊆ 𝑡)) | |
5 | 3, 4 | anbi12d 634 | . . . . . . 7 ⊢ (𝑠 = 𝑡 → ((∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠) ↔ (∪ 𝑡 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑡))) |
6 | 5 | elrab 3591 | . . . . . 6 ⊢ (𝑡 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ↔ (𝑡 ∈ SAlg ∧ (∪ 𝑡 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑡))) |
7 | 6 | biimpi 219 | . . . . 5 ⊢ (𝑡 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} → (𝑡 ∈ SAlg ∧ (∪ 𝑡 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑡))) |
8 | 7 | simprrd 774 | . . . 4 ⊢ (𝑡 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} → 𝑋 ⊆ 𝑡) |
9 | 1, 8 | mprgbir 3066 | . . 3 ⊢ 𝑋 ⊆ ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} |
10 | 9 | a1i 11 | . 2 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ⊆ ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)}) |
11 | sssalgen.1 | . . 3 ⊢ 𝑆 = (SalGen‘𝑋) | |
12 | salgenval 43480 | . . 3 ⊢ (𝑋 ∈ 𝑉 → (SalGen‘𝑋) = ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)}) | |
13 | 11, 12 | eqtr2id 2784 | . 2 ⊢ (𝑋 ∈ 𝑉 → ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} = 𝑆) |
14 | 10, 13 | sseqtrd 3927 | 1 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ⊆ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 {crab 3055 ⊆ wss 3853 ∪ cuni 4805 ∩ cint 4845 ‘cfv 6358 SAlgcsalg 43467 SalGencsalgen 43471 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-int 4846 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-iota 6316 df-fun 6360 df-fv 6366 df-salg 43468 df-salgen 43472 |
This theorem is referenced by: dfsalgen2 43498 iooborel 43508 opnssborel 43791 cnfsmf 43891 |
Copyright terms: Public domain | W3C validator |