Theorem salgenuni 42977

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 42963	. . . . 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 2833	. . 3 ⊢ (𝜑 → 𝑆 = ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)})
7	6	unieqd 4814	. 2 ⊢ (𝜑 → ∪ 𝑆 = ∪ ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)})
8		ssrab2 4007	. . . 4 ⊢ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ⊆ SAlg
9	8	a1i 11	. . 3 ⊢ (𝜑 → {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ⊆ SAlg)
10		salgenn0 42971	. . . 4 ⊢ (𝑋 ∈ 𝑉 → {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ≠ ∅)
11	3, 10	syl 17	. . 3 ⊢ (𝜑 → {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ≠ ∅)
12		unieq 4811	. . . . . . . . . 10 ⊢ (𝑠 = 𝑡 → ∪ 𝑠 = ∪ 𝑡)
13	12	eqeq1d 2800	. . . . . . . . 9 ⊢ (𝑠 = 𝑡 → (∪ 𝑠 = ∪ 𝑋 ↔ ∪ 𝑡 = ∪ 𝑋))
14		sseq2 3941	. . . . . . . . 9 ⊢ (𝑠 = 𝑡 → (𝑋 ⊆ 𝑠 ↔ 𝑋 ⊆ 𝑡))
15	13, 14	anbi12d 633	. . . . . . . 8 ⊢ (𝑠 = 𝑡 → ((∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠) ↔ (∪ 𝑡 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑡)))
16	15	elrab 3628	. . . . . . 7 ⊢ (𝑡 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ↔ (𝑡 ∈ SAlg ∧ (∪ 𝑡 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑡)))
17	16	biimpi 219	. . . . . 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 2807	. . . . . 6 ⊢ ∪ 𝑋 = 𝑈
21	20	a1i 11	. . . . 5 ⊢ (𝑡 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} → ∪ 𝑋 = 𝑈)
22	18, 21	eqtrd 2833	. . . 4 ⊢ (𝑡 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} → ∪ 𝑡 = 𝑈)
23	22	adantl 485	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)}) → ∪ 𝑡 = 𝑈)
24	9, 11, 23	intsaluni 42969	. 2 ⊢ (𝜑 → ∪ ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} = 𝑈)
25	7, 24	eqtrd 2833	1 ⊢ (𝜑 → ∪ 𝑆 = 𝑈)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  {crab 3110   ⊆ wss 3881  ∅c0 4243  ∪ cuni 4800  ∩ cint 4838  ‘cfv 6324  SAlgcsalg 42950  SalGencsalgen 42954

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-int 4839  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-salg 42951  df-salgen 42955

This theorem is referenced by:  unisalgen  42980  dfsalgen2  42981