Theorem sigaclcuni 34517

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 4994	. . 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 3128	. . . . . . . 8 ⊢ ((∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵) → ∃𝑘 ∈ 𝐴 (𝐵 ∈ 𝑆 ∧ 𝑧 = 𝐵))
5		simpr 489	. . . . . . . . . 10 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝑧 = 𝐵) → 𝑧 = 𝐵)
6		simpl 487	. . . . . . . . . 10 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝑧 = 𝐵) → 𝐵 ∈ 𝑆)
7	5, 6	eqeltrd 2863	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝑧 = 𝐵) → 𝑧 ∈ 𝑆)
8	7	rexlimivw 3162	. . . . . . . 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 417	. . . . . 6 ⊢ (∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 → (∃𝑘 ∈ 𝐴 𝑧 = 𝐵 → 𝑧 ∈ 𝑆))
11	10	abssdv 4021	. . . . 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 5304	. . . . 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 33071	. . . 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 34516	. . 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 2863	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 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  {cab 2741  Ⅎwnfc 2910  ∀wral 3079  ∃wrex 3089   ⊆ wss 3905  𝒫 cpw 4562  ∪ cuni 4872  ∪ ciun 4956   class class class wbr 5109  ran crn 5662  ωcom 7858   ≼ cdom 8937  sigAlgebracsiga 34507

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-inf2 9606  ax-ac2 10451

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-er 8690  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-oi 9468  df-card 9930  df-acn 9933  df-ac 10105  df-siga 34508

This theorem is used by:  measvuni  34613  imambfm  34661  sibfof  34739