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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 2747 | . . . . 5 ⊢ (SalGen‘𝐴) = 𝑆 |
3 | 2 | a1i 11 | . . . 4 ⊢ (𝜑 → (SalGen‘𝐴) = 𝑆) |
4 | unisalgen2.x | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
5 | 4 | dfsalgen2 43880 | . . . 4 ⊢ (𝜑 → ((SalGen‘𝐴) = 𝑆 ↔ ((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝐴 ∧ 𝐴 ⊆ 𝑆) ∧ ∀𝑥 ∈ SAlg ((∪ 𝑥 = ∪ 𝐴 ∧ 𝐴 ⊆ 𝑥) → 𝑆 ⊆ 𝑥)))) |
6 | 3, 5 | mpbid 231 | . . 3 ⊢ (𝜑 → ((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝐴 ∧ 𝐴 ⊆ 𝑆) ∧ ∀𝑥 ∈ SAlg ((∪ 𝑥 = ∪ 𝐴 ∧ 𝐴 ⊆ 𝑥) → 𝑆 ⊆ 𝑥))) |
7 | 6 | simpld 495 | . 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 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ⊆ wss 3887 ∪ cuni 4839 ‘cfv 6433 SAlgcsalg 43849 SalGencsalgen 43853 |
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 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 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 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-iota 6391 df-fun 6435 df-fv 6441 df-salg 43850 df-salgen 43854 |
This theorem is referenced by: incsmf 44278 decsmf 44302 |
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