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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 2746 | . . . . 5 ⊢ (SalGen‘𝐴) = 𝑆 |
| 3 | 2 | a1i 11 | . . . 4 ⊢ (𝜑 → (SalGen‘𝐴) = 𝑆) |
| 4 | unisalgen2.x | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
| 5 | 4 | dfsalgen2 46356 | . . . 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 1144 | 1 ⊢ (𝜑 → ∪ 𝑆 = ∪ 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ⊆ wss 3951 ∪ cuni 4907 ‘cfv 6561 SAlgcsalg 46323 SalGencsalgen 46327 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-salg 46324 df-salgen 46328 |
| This theorem is referenced by: incsmf 46757 decsmf 46782 |
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