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Mirrors > Home > MPE Home > Th. List > Mathboxes > salgencld | Structured version Visualization version GIF version |
Description: SalGen actually generates a sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
salgencld.1 | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
salgencld.2 | ⊢ 𝑆 = (SalGen‘𝑋) |
Ref | Expression |
---|---|
salgencld | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | salgencld.2 | . 2 ⊢ 𝑆 = (SalGen‘𝑋) | |
2 | salgencld.1 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
3 | salgencl 43841 | . . 3 ⊢ (𝑋 ∈ 𝑉 → (SalGen‘𝑋) ∈ SAlg) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (𝜑 → (SalGen‘𝑋) ∈ SAlg) |
5 | 1, 4 | eqeltrid 2843 | 1 ⊢ (𝜑 → 𝑆 ∈ SAlg) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 ‘cfv 6435 SAlgcsalg 43819 SalGencsalgen 43823 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5225 ax-nul 5232 ax-pow 5290 ax-pr 5354 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-int 4882 df-br 5077 df-opab 5139 df-mpt 5160 df-id 5491 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-iota 6393 df-fun 6437 df-fv 6443 df-salg 43820 df-salgen 43824 |
This theorem is referenced by: bor1sal 43864 cnfsmf 44243 incsmf 44245 bormflebmf 44256 decsmf 44269 smf2id 44302 smfco 44303 |
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