Theorem sigaclcuni 34732

Description: A sigma-algebra is closed under countable union: indexed union version. (Contributed by Thierry Arnoux, 8-Jun-2017.)

Hypothesis

Ref	Expression
sigaclcuni.1	⊢ Ⅎ𝑘𝐴

Assertion

Ref	Expression
sigaclcuni	⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ 𝐴 ≼ ω) → ∪ 𝑘 ∈ 𝐴 𝐵 ∈ 𝑆)

Distinct variable group:   𝑆,𝑘

Allowed substitution hints:   𝐴(𝑘)   𝐵(𝑘)

Proof of Theorem sigaclcuni

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfiun2g 4988	. . 3 ⊢ (∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 → ∪ 𝑘 ∈ 𝐴 𝐵 = ∪ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵})
2	1	3ad2ant2 1152	. 2 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ 𝐴 ≼ ω) → ∪ 𝑘 ∈ 𝐴 𝐵 = ∪ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵})
3		simp1 1154	. . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ 𝐴 ≼ ω) → 𝑆 ∈ ∪ ran sigAlgebra)
4		r19.29 3126	. . . . . . . 8 ⊢ ((∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵) → ∃𝑘 ∈ 𝐴 (𝐵 ∈ 𝑆 ∧ 𝑧 = 𝐵))
5		simpr 490	. . . . . . . . . 10 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝑧 = 𝐵) → 𝑧 = 𝐵)
6		simpl 488	. . . . . . . . . 10 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝑧 = 𝐵) → 𝐵 ∈ 𝑆)
7	5, 6	eqeltrd 2861	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝑧 = 𝐵) → 𝑧 ∈ 𝑆)
8	7	rexlimivw 3160	. . . . . . . 8 ⊢ (∃𝑘 ∈ 𝐴 (𝐵 ∈ 𝑆 ∧ 𝑧 = 𝐵) → 𝑧 ∈ 𝑆)
9	4, 8	syl 18	. . . . . . 7 ⊢ ((∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵) → 𝑧 ∈ 𝑆)
10	9	ex 418	. . . . . 6 ⊢ (∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 → (∃𝑘 ∈ 𝐴 𝑧 = 𝐵 → 𝑧 ∈ 𝑆))
11	10	abssdv 4015	. . . . 5 ⊢ (∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 → {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵} ⊆ 𝑆)
12	11	3ad2ant2 1152	. . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ 𝐴 ≼ ω) → {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵} ⊆ 𝑆)
13		elpw2g 5295	. . . . 5 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → ({𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵} ∈ 𝒫 𝑆 ↔ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵} ⊆ 𝑆))
14	3, 13	syl 18	. . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ 𝐴 ≼ ω) → ({𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵} ∈ 𝒫 𝑆 ↔ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵} ⊆ 𝑆))
15	12, 14	mpbird 260	. . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ 𝐴 ≼ ω) → {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵} ∈ 𝒫 𝑆)
16		sigaclcuni.1	. . . . 5 ⊢ Ⅎ𝑘𝐴
17	16	abrexctf 33291	. . . 4 ⊢ (𝐴 ≼ ω → {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵} ≼ ω)
18	17	3ad2ant3 1153	. . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ 𝐴 ≼ ω) → {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵} ≼ ω)
19		sigaclcu 34731	. . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵} ∈ 𝒫 𝑆 ∧ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵} ≼ ω) → ∪ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵} ∈ 𝑆)
20	3, 15, 18, 19	syl3anc 1398	. 2 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ 𝐴 ≼ ω) → ∪ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵} ∈ 𝑆)
21	2, 20	eqeltrd 2861	1 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ 𝐴 ≼ ω) → ∪ 𝑘 ∈ 𝐴 𝐵 ∈ 𝑆)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  {cab 2739  Ⅎwnfc 2908  ∀wral 3077  ∃wrex 3087   ⊆ wss 3899  𝒫 cpw 4557  ∪ cuni 4867  ∪ ciun 4951   class class class wbr 5103  ran crn 5652  ωcom 7866   ≼ cdom 8955  sigAlgebracsiga 34722

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-inf2 9626

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-isom 6540  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8460  df-er 8701  df-map 8833  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-oi 9488  df-card 10001  df-acn 10004  df-siga 34723

This theorem is used by:  measvuni  34829  imambfm  34877  sibfof  34955