Theorem salgenuni 43766

Description: The base set of the sigma-algebra generated by a set is the union of the set itself. (Contributed by Glauco Siliprandi, 3-Jan-2021.)

Hypotheses

Ref	Expression
salgenuni.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
salgenuni.s	⊢ 𝑆 = (SalGen‘𝑋)
salgenuni.u	⊢ 𝑈 = ∪ 𝑋

Assertion

Ref	Expression
salgenuni	⊢ (𝜑 → ∪ 𝑆 = 𝑈)

Proof of Theorem salgenuni

Dummy variables 𝑡 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		salgenuni.s	. . . . 5 ⊢ 𝑆 = (SalGen‘𝑋)
2	1	a1i 11	. . . 4 ⊢ (𝜑 → 𝑆 = (SalGen‘𝑋))
3		salgenuni.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
4		salgenval 43752	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (SalGen‘𝑋) = ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)})
5	3, 4	syl 17	. . . 4 ⊢ (𝜑 → (SalGen‘𝑋) = ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)})
6	2, 5	eqtrd 2778	. . 3 ⊢ (𝜑 → 𝑆 = ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)})
7	6	unieqd 4850	. 2 ⊢ (𝜑 → ∪ 𝑆 = ∪ ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)})
8		ssrab2 4009	. . . 4 ⊢ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ⊆ SAlg
9	8	a1i 11	. . 3 ⊢ (𝜑 → {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ⊆ SAlg)
10		salgenn0 43760	. . . 4 ⊢ (𝑋 ∈ 𝑉 → {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ≠ ∅)
11	3, 10	syl 17	. . 3 ⊢ (𝜑 → {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ≠ ∅)
12		unieq 4847	. . . . . . . . . 10 ⊢ (𝑠 = 𝑡 → ∪ 𝑠 = ∪ 𝑡)
13	12	eqeq1d 2740	. . . . . . . . 9 ⊢ (𝑠 = 𝑡 → (∪ 𝑠 = ∪ 𝑋 ↔ ∪ 𝑡 = ∪ 𝑋))
14		sseq2 3943	. . . . . . . . 9 ⊢ (𝑠 = 𝑡 → (𝑋 ⊆ 𝑠 ↔ 𝑋 ⊆ 𝑡))
15	13, 14	anbi12d 630	. . . . . . . 8 ⊢ (𝑠 = 𝑡 → ((∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠) ↔ (∪ 𝑡 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑡)))
16	15	elrab 3617	. . . . . . 7 ⊢ (𝑡 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ↔ (𝑡 ∈ SAlg ∧ (∪ 𝑡 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑡)))
17	16	biimpi 215	. . . . . 6 ⊢ (𝑡 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} → (𝑡 ∈ SAlg ∧ (∪ 𝑡 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑡)))
18	17	simprld 768	. . . . 5 ⊢ (𝑡 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} → ∪ 𝑡 = ∪ 𝑋)
19		salgenuni.u	. . . . . . 7 ⊢ 𝑈 = ∪ 𝑋
20	19	eqcomi 2747	. . . . . 6 ⊢ ∪ 𝑋 = 𝑈
21	20	a1i 11	. . . . 5 ⊢ (𝑡 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} → ∪ 𝑋 = 𝑈)
22	18, 21	eqtrd 2778	. . . 4 ⊢ (𝑡 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} → ∪ 𝑡 = 𝑈)
23	22	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)}) → ∪ 𝑡 = 𝑈)
24	9, 11, 23	intsaluni 43758	. 2 ⊢ (𝜑 → ∪ ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} = 𝑈)
25	7, 24	eqtrd 2778	1 ⊢ (𝜑 → ∪ 𝑆 = 𝑈)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  {crab 3067   ⊆ wss 3883  ∅c0 4253  ∪ cuni 4836  ∩ cint 4876  ‘cfv 6418  SAlgcsalg 43739  SalGencsalgen 43743

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-int 4877  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-iota 6376  df-fun 6420  df-fv 6426  df-salg 43740  df-salgen 43744

This theorem is referenced by:  unisalgen  43769  dfsalgen2  43770