Theorem sigaclcuni 32114

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 4963	. . 3 ⊢ (∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 → ∪ 𝑘 ∈ 𝐴 𝐵 = ∪ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵})
2	1	3ad2ant2 1132	. 2 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ 𝐴 ≼ ω) → ∪ 𝑘 ∈ 𝐴 𝐵 = ∪ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵})
3		simp1 1134	. . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ 𝐴 ≼ ω) → 𝑆 ∈ ∪ ran sigAlgebra)
4		r19.29 3111	. . . . . . . 8 ⊢ ((∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵) → ∃𝑘 ∈ 𝐴 (𝐵 ∈ 𝑆 ∧ 𝑧 = 𝐵))
5		simpr 484	. . . . . . . . . 10 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝑧 = 𝐵) → 𝑧 = 𝐵)
6		simpl 482	. . . . . . . . . 10 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝑧 = 𝐵) → 𝐵 ∈ 𝑆)
7	5, 6	eqeltrd 2834	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝑧 = 𝐵) → 𝑧 ∈ 𝑆)
8	7	rexlimivw 3142	. . . . . . . 8 ⊢ (∃𝑘 ∈ 𝐴 (𝐵 ∈ 𝑆 ∧ 𝑧 = 𝐵) → 𝑧 ∈ 𝑆)
9	4, 8	syl 17	. . . . . . 7 ⊢ ((∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵) → 𝑧 ∈ 𝑆)
10	9	ex 412	. . . . . 6 ⊢ (∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 → (∃𝑘 ∈ 𝐴 𝑧 = 𝐵 → 𝑧 ∈ 𝑆))
11	10	abssdv 4004	. . . . 5 ⊢ (∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 → {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵} ⊆ 𝑆)
12	11	3ad2ant2 1132	. . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ 𝐴 ≼ ω) → {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵} ⊆ 𝑆)
13		elpw2g 5271	. . . . 5 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → ({𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵} ∈ 𝒫 𝑆 ↔ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵} ⊆ 𝑆))
14	3, 13	syl 17	. . . 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 256	. . 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 31081	. . . 4 ⊢ (𝐴 ≼ ω → {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵} ≼ ω)
18	17	3ad2ant3 1133	. . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ 𝐴 ≼ ω) → {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵} ≼ ω)
19		sigaclcu 32113	. . 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 1369	. 2 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ 𝐴 ≼ ω) → ∪ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵} ∈ 𝑆)
21	2, 20	eqeltrd 2834	1 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ 𝐴 ≼ ω) → ∪ 𝑘 ∈ 𝐴 𝐵 ∈ 𝑆)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1537   ∈ wcel 2101  {cab 2710  Ⅎwnfc 2882  ∀wral 3059  ∃wrex 3068   ⊆ wss 3889  𝒫 cpw 4536  ∪ cuni 4841  ∪ ciun 4927   class class class wbr 5077  ran crn 5592  ωcom 7732   ≼ cdom 8751  sigAlgebracsiga 32104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-10 2132  ax-11 2149  ax-12 2166  ax-ext 2704  ax-rep 5212  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7608  ax-inf2 9427  ax-ac2 10247

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2063  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2884  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3222  df-reu 3223  df-rab 3224  df-v 3436  df-sbc 3719  df-csb 3835  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3908  df-nul 4260  df-if 4463  df-pw 4538  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4842  df-int 4883  df-iun 4929  df-br 5078  df-opab 5140  df-mpt 5161  df-tr 5195  df-id 5491  df-eprel 5497  df-po 5505  df-so 5506  df-fr 5546  df-se 5547  df-we 5548  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-rn 5602  df-res 5603  df-ima 5604  df-pred 6206  df-ord 6273  df-on 6274  df-lim 6275  df-suc 6276  df-iota 6399  df-fun 6449  df-fn 6450  df-f 6451  df-f1 6452  df-fo 6453  df-f1o 6454  df-fv 6455  df-isom 6456  df-riota 7252  df-ov 7298  df-oprab 7299  df-mpo 7300  df-om 7733  df-1st 7851  df-2nd 7852  df-frecs 8117  df-wrecs 8148  df-recs 8222  df-rdg 8261  df-1o 8317  df-er 8518  df-map 8637  df-en 8754  df-dom 8755  df-sdom 8756  df-fin 8757  df-oi 9297  df-card 9725  df-acn 9728  df-ac 9900  df-siga 32105

This theorem is referenced by:  measvuni  32210  imambfm  32257  sibfof  32335