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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sssigagen2 | Structured version Visualization version GIF version |
Description: A subset of the generating set is also a subset of the generated sigma-algebra. (Contributed by Thierry Arnoux, 22-Sep-2017.) |
Ref | Expression |
---|---|
sssigagen2 | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ (sigaGen‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 483 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ 𝐴) | |
2 | sssigagen 33934 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (sigaGen‘𝐴)) | |
3 | 2 | adantr 479 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴) → 𝐴 ⊆ (sigaGen‘𝐴)) |
4 | 1, 3 | sstrd 3989 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ (sigaGen‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2098 ⊆ wss 3946 ‘cfv 6553 sigaGencsigagen 33927 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5579 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-iota 6505 df-fun 6555 df-fv 6561 df-siga 33898 df-sigagen 33928 |
This theorem is referenced by: sxbrsigalem5 34078 |
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