Theorem sigaclcuni 33116

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 5034	. . 3 ⊢ (∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 → ∪ 𝑘 ∈ 𝐴 𝐵 = ∪ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵})
2	1	3ad2ant2 1135	. 2 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ 𝐴 ≼ ω) → ∪ 𝑘 ∈ 𝐴 𝐵 = ∪ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵})
3		simp1 1137	. . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ 𝐴 ≼ ω) → 𝑆 ∈ ∪ ran sigAlgebra)
4		r19.29 3115	. . . . . . . 8 ⊢ ((∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵) → ∃𝑘 ∈ 𝐴 (𝐵 ∈ 𝑆 ∧ 𝑧 = 𝐵))
5		simpr 486	. . . . . . . . . 10 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝑧 = 𝐵) → 𝑧 = 𝐵)
6		simpl 484	. . . . . . . . . 10 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝑧 = 𝐵) → 𝐵 ∈ 𝑆)
7	5, 6	eqeltrd 2834	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝑧 = 𝐵) → 𝑧 ∈ 𝑆)
8	7	rexlimivw 3152	. . . . . . . 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 414	. . . . . 6 ⊢ (∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 → (∃𝑘 ∈ 𝐴 𝑧 = 𝐵 → 𝑧 ∈ 𝑆))
11	10	abssdv 4066	. . . . 5 ⊢ (∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 → {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵} ⊆ 𝑆)
12	11	3ad2ant2 1135	. . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ 𝐴 ≼ ω) → {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵} ⊆ 𝑆)
13		elpw2g 5345	. . . . 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 257	. . 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 31943	. . . 4 ⊢ (𝐴 ≼ ω → {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵} ≼ ω)
18	17	3ad2ant3 1136	. . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ 𝐴 ≼ ω) → {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵} ≼ ω)
19		sigaclcu 33115	. . 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 1372	. 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  {cab 2710  Ⅎwnfc 2884  ∀wral 3062  ∃wrex 3071   ⊆ wss 3949  𝒫 cpw 4603  ∪ cuni 4909  ∪ ciun 4998   class class class wbr 5149  ran crn 5678  ωcom 7855   ≼ cdom 8937  sigAlgebracsiga 33106

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636  ax-ac2 10458

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-oi 9505  df-card 9934  df-acn 9937  df-ac 10111  df-siga 33107

This theorem is referenced by:  measvuni  33212  imambfm  33261  sibfof  33339