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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 42622 | . . 3 ⊢ (𝑋 ∈ 𝑉 → (SalGen‘𝑋) ∈ SAlg) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (𝜑 → (SalGen‘𝑋) ∈ SAlg) |
5 | 1, 4 | eqeltrid 2919 | 1 ⊢ (𝜑 → 𝑆 ∈ SAlg) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ‘cfv 6357 SAlgcsalg 42600 SalGencsalgen 42604 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-int 4879 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-salg 42601 df-salgen 42605 |
This theorem is referenced by: bor1sal 42645 cnfsmf 43024 incsmf 43026 bormflebmf 43037 decsmf 43050 smf2id 43083 smfco 43084 |
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