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Mathbox for Glauco Siliprandi |
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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 41341 | . . 3 ⊢ (𝑋 ∈ 𝑉 → (SalGen‘𝑋) ∈ SAlg) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (𝜑 → (SalGen‘𝑋) ∈ SAlg) |
5 | 1, 4 | syl5eqel 2910 | 1 ⊢ (𝜑 → 𝑆 ∈ SAlg) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 ‘cfv 6123 SAlgcsalg 41319 SalGencsalgen 41323 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-int 4698 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-iota 6086 df-fun 6125 df-fv 6131 df-salg 41320 df-salgen 41324 |
This theorem is referenced by: bor1sal 41364 cnfsmf 41743 incsmf 41745 bormflebmf 41756 decsmf 41769 smf2id 41802 smfco 41803 |
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