Theorem sigaclcuni 30181

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 4552	. . 3 ⊢ (∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 → ∪ 𝑘 ∈ 𝐴 𝐵 = ∪ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵})
2	1	3ad2ant2 1083	. 2 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ 𝐴 ≼ ω) → ∪ 𝑘 ∈ 𝐴 𝐵 = ∪ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵})
3		simp1 1061	. . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ 𝐴 ≼ ω) → 𝑆 ∈ ∪ ran sigAlgebra)
4		r19.29 3072	. . . . . . . 8 ⊢ ((∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵) → ∃𝑘 ∈ 𝐴 (𝐵 ∈ 𝑆 ∧ 𝑧 = 𝐵))
5		simpr 477	. . . . . . . . . 10 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝑧 = 𝐵) → 𝑧 = 𝐵)
6		simpl 473	. . . . . . . . . 10 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝑧 = 𝐵) → 𝐵 ∈ 𝑆)
7	5, 6	eqeltrd 2701	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝑧 = 𝐵) → 𝑧 ∈ 𝑆)
8	7	rexlimivw 3029	. . . . . . . 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 450	. . . . . 6 ⊢ (∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 → (∃𝑘 ∈ 𝐴 𝑧 = 𝐵 → 𝑧 ∈ 𝑆))
11	10	abssdv 3676	. . . . 5 ⊢ (∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 → {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵} ⊆ 𝑆)
12	11	3ad2ant2 1083	. . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ 𝐴 ≼ ω) → {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵} ⊆ 𝑆)
13		elpw2g 4827	. . . . 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 247	. . 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 29496	. . . 4 ⊢ (𝐴 ≼ ω → {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵} ≼ ω)
18	17	3ad2ant3 1084	. . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ 𝐴 ≼ ω) → {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵} ≼ ω)
19		sigaclcu 30180	. . 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 1326	. 2 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ 𝐴 ≼ ω) → ∪ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵} ∈ 𝑆)
21	2, 20	eqeltrd 2701	1 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ 𝐴 ≼ ω) → ∪ 𝑘 ∈ 𝐴 𝐵 ∈ 𝑆)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  {cab 2608  Ⅎwnfc 2751  ∀wral 2912  ∃wrex 2913   ⊆ wss 3574  𝒫 cpw 4158  ∪ cuni 4436  ∪ ciun 4520   class class class wbr 4653  ran crn 5115  ωcom 7065   ≼ cdom 7953  sigAlgebracsiga 30170

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-ac2 9285

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-oi 8415  df-card 8765  df-acn 8768  df-ac 8939  df-siga 30171

This theorem is referenced by:  measvuni  30277  imambfm  30324  sibfof  30402