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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 2631 | . . . . 5 ⊢ (SalGen‘𝐴) = 𝑆 |
3 | 2 | a1i 11 | . . . 4 ⊢ (𝜑 → (SalGen‘𝐴) = 𝑆) |
4 | unisalgen2.x | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
5 | 4 | dfsalgen2 40559 | . . . 4 ⊢ (𝜑 → ((SalGen‘𝐴) = 𝑆 ↔ ((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝐴 ∧ 𝐴 ⊆ 𝑆) ∧ ∀𝑥 ∈ SAlg ((∪ 𝑥 = ∪ 𝐴 ∧ 𝐴 ⊆ 𝑥) → 𝑆 ⊆ 𝑥)))) |
6 | 3, 5 | mpbid 222 | . . 3 ⊢ (𝜑 → ((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝐴 ∧ 𝐴 ⊆ 𝑆) ∧ ∀𝑥 ∈ SAlg ((∪ 𝑥 = ∪ 𝐴 ∧ 𝐴 ⊆ 𝑥) → 𝑆 ⊆ 𝑥))) |
7 | 6 | simpld 475 | . 2 ⊢ (𝜑 → (𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝐴 ∧ 𝐴 ⊆ 𝑆)) |
8 | 7 | simp2d 1074 | 1 ⊢ (𝜑 → ∪ 𝑆 = ∪ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ⊆ wss 3574 ∪ cuni 4436 ‘cfv 5888 SAlgcsalg 40528 SalGencsalgen 40532 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-int 4476 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-salg 40529 df-salgen 40533 |
This theorem is referenced by: incsmf 40951 decsmf 40975 |
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