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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > unisalgen2 | Structured version Visualization version GIF version |
Description: The union of a set belongs is equal to the union of the sigma-algebra generated by the set. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
unisalgen2.x | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
unisalgen2.s | ⊢ 𝑆 = (SalGen‘𝐴) |
Ref | Expression |
---|---|
unisalgen2 | ⊢ (𝜑 → ∪ 𝑆 = ∪ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unisalgen2.s | . . . . . 6 ⊢ 𝑆 = (SalGen‘𝐴) | |
2 | 1 | eqcomi 2743 | . . . . 5 ⊢ (SalGen‘𝐴) = 𝑆 |
3 | 2 | a1i 11 | . . . 4 ⊢ (𝜑 → (SalGen‘𝐴) = 𝑆) |
4 | unisalgen2.x | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
5 | 4 | dfsalgen2 46296 | . . . 4 ⊢ (𝜑 → ((SalGen‘𝐴) = 𝑆 ↔ ((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝐴 ∧ 𝐴 ⊆ 𝑆) ∧ ∀𝑥 ∈ SAlg ((∪ 𝑥 = ∪ 𝐴 ∧ 𝐴 ⊆ 𝑥) → 𝑆 ⊆ 𝑥)))) |
6 | 3, 5 | mpbid 232 | . . 3 ⊢ (𝜑 → ((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝐴 ∧ 𝐴 ⊆ 𝑆) ∧ ∀𝑥 ∈ SAlg ((∪ 𝑥 = ∪ 𝐴 ∧ 𝐴 ⊆ 𝑥) → 𝑆 ⊆ 𝑥))) |
7 | 6 | simpld 494 | . 2 ⊢ (𝜑 → (𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝐴 ∧ 𝐴 ⊆ 𝑆)) |
8 | 7 | simp2d 1142 | 1 ⊢ (𝜑 → ∪ 𝑆 = ∪ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ∀wral 3058 ⊆ wss 3962 ∪ cuni 4911 ‘cfv 6562 SAlgcsalg 46263 SalGencsalgen 46267 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-iota 6515 df-fun 6564 df-fv 6570 df-salg 46264 df-salgen 46268 |
This theorem is referenced by: incsmf 46697 decsmf 46722 |
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