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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 43442 | . . . 4 ⊢ (𝜑 → ((SalGen‘𝐴) = 𝑆 ↔ ((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝐴 ∧ 𝐴 ⊆ 𝑆) ∧ ∀𝑥 ∈ SAlg ((∪ 𝑥 = ∪ 𝐴 ∧ 𝐴 ⊆ 𝑥) → 𝑆 ⊆ 𝑥)))) |
6 | 3, 5 | mpbid 235 | . . 3 ⊢ (𝜑 → ((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝐴 ∧ 𝐴 ⊆ 𝑆) ∧ ∀𝑥 ∈ SAlg ((∪ 𝑥 = ∪ 𝐴 ∧ 𝐴 ⊆ 𝑥) → 𝑆 ⊆ 𝑥))) |
7 | 6 | simpld 498 | . 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 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ∀wral 3053 ⊆ wss 3843 ∪ cuni 4796 ‘cfv 6339 SAlgcsalg 43411 SalGencsalgen 43415 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-int 4837 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-iota 6297 df-fun 6341 df-fv 6347 df-salg 43412 df-salgen 43416 |
This theorem is referenced by: incsmf 43837 decsmf 43861 |
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