Theorem salgenuni 46342

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 46326	. . . . 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 2765	. . 3 ⊢ (𝜑 → 𝑆 = ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)})
7	6	unieqd 4887	. 2 ⊢ (𝜑 → ∪ 𝑆 = ∪ ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)})
8		ssrab2 4046	. . . 4 ⊢ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ⊆ SAlg
9	8	a1i 11	. . 3 ⊢ (𝜑 → {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ⊆ SAlg)
10		salgenn0 46336	. . . 4 ⊢ (𝑋 ∈ 𝑉 → {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ≠ ∅)
11	3, 10	syl 17	. . 3 ⊢ (𝜑 → {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ≠ ∅)
12		unieq 4885	. . . . . . . . . 10 ⊢ (𝑠 = 𝑡 → ∪ 𝑠 = ∪ 𝑡)
13	12	eqeq1d 2732	. . . . . . . . 9 ⊢ (𝑠 = 𝑡 → (∪ 𝑠 = ∪ 𝑋 ↔ ∪ 𝑡 = ∪ 𝑋))
14		sseq2 3976	. . . . . . . . 9 ⊢ (𝑠 = 𝑡 → (𝑋 ⊆ 𝑠 ↔ 𝑋 ⊆ 𝑡))
15	13, 14	anbi12d 632	. . . . . . . 8 ⊢ (𝑠 = 𝑡 → ((∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠) ↔ (∪ 𝑡 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑡)))
16	15	elrab 3662	. . . . . . 7 ⊢ (𝑡 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ↔ (𝑡 ∈ SAlg ∧ (∪ 𝑡 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑡)))
17	16	biimpi 216	. . . . . 6 ⊢ (𝑡 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} → (𝑡 ∈ SAlg ∧ (∪ 𝑡 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑡)))
18	17	simprld 771	. . . . 5 ⊢ (𝑡 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} → ∪ 𝑡 = ∪ 𝑋)
19		salgenuni.u	. . . . . . 7 ⊢ 𝑈 = ∪ 𝑋
20	19	eqcomi 2739	. . . . . 6 ⊢ ∪ 𝑋 = 𝑈
21	20	a1i 11	. . . . 5 ⊢ (𝑡 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} → ∪ 𝑋 = 𝑈)
22	18, 21	eqtrd 2765	. . . 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 46334	. 2 ⊢ (𝜑 → ∪ ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} = 𝑈)
25	7, 24	eqtrd 2765	1 ⊢ (𝜑 → ∪ 𝑆 = 𝑈)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  {crab 3408   ⊆ wss 3917  ∅c0 4299  ∪ cuni 4874  ∩ cint 4913  ‘cfv 6514  SAlgcsalg 46313  SalGencsalgen 46317

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-iota 6467  df-fun 6516  df-fv 6522  df-salg 46314  df-salgen 46318

This theorem is referenced by:  unisalgen  46345  dfsalgen2  46346