Theorem saliuncl 40542

Description: SAlg sigma-algebra is closed under countable indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
saliuncl.s	⊢ (𝜑 → 𝑆 ∈ SAlg)
saliuncl.kct	⊢ (𝜑 → 𝐾 ≼ ω)
saliuncl.b	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐾) → 𝐸 ∈ 𝑆)

Assertion

Ref	Expression
saliuncl	⊢ (𝜑 → ∪ 𝑘 ∈ 𝐾 𝐸 ∈ 𝑆)

Distinct variable groups:   𝑘,𝐾   𝑆,𝑘   𝜑,𝑘

Allowed substitution hint:   𝐸(𝑘)

Proof of Theorem saliuncl

Step	Hyp	Ref	Expression
1		saliuncl.b	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐾) → 𝐸 ∈ 𝑆)
2	1	ralrimiva 2966	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ 𝐾 𝐸 ∈ 𝑆)
3		dfiun3g 5378	. . 3 ⊢ (∀𝑘 ∈ 𝐾 𝐸 ∈ 𝑆 → ∪ 𝑘 ∈ 𝐾 𝐸 = ∪ ran (𝑘 ∈ 𝐾 ↦ 𝐸))
4	2, 3	syl 17	. 2 ⊢ (𝜑 → ∪ 𝑘 ∈ 𝐾 𝐸 = ∪ ran (𝑘 ∈ 𝐾 ↦ 𝐸))
5		saliuncl.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg)
6		eqid 2622	. . . . . 6 ⊢ (𝑘 ∈ 𝐾 ↦ 𝐸) = (𝑘 ∈ 𝐾 ↦ 𝐸)
7	6	rnmptss 6392	. . . . 5 ⊢ (∀𝑘 ∈ 𝐾 𝐸 ∈ 𝑆 → ran (𝑘 ∈ 𝐾 ↦ 𝐸) ⊆ 𝑆)
8	2, 7	syl 17	. . . 4 ⊢ (𝜑 → ran (𝑘 ∈ 𝐾 ↦ 𝐸) ⊆ 𝑆)
9	5, 8	ssexd 4805	. . . . 5 ⊢ (𝜑 → ran (𝑘 ∈ 𝐾 ↦ 𝐸) ∈ V)
10		elpwg 4166	. . . . 5 ⊢ (ran (𝑘 ∈ 𝐾 ↦ 𝐸) ∈ V → (ran (𝑘 ∈ 𝐾 ↦ 𝐸) ∈ 𝒫 𝑆 ↔ ran (𝑘 ∈ 𝐾 ↦ 𝐸) ⊆ 𝑆))
11	9, 10	syl 17	. . . 4 ⊢ (𝜑 → (ran (𝑘 ∈ 𝐾 ↦ 𝐸) ∈ 𝒫 𝑆 ↔ ran (𝑘 ∈ 𝐾 ↦ 𝐸) ⊆ 𝑆))
12	8, 11	mpbird 247	. . 3 ⊢ (𝜑 → ran (𝑘 ∈ 𝐾 ↦ 𝐸) ∈ 𝒫 𝑆)
13		saliuncl.kct	. . . 4 ⊢ (𝜑 → 𝐾 ≼ ω)
14		1stcrestlem 21255	. . . 4 ⊢ (𝐾 ≼ ω → ran (𝑘 ∈ 𝐾 ↦ 𝐸) ≼ ω)
15	13, 14	syl 17	. . 3 ⊢ (𝜑 → ran (𝑘 ∈ 𝐾 ↦ 𝐸) ≼ ω)
16	5, 12, 15	salunicl 40536	. 2 ⊢ (𝜑 → ∪ ran (𝑘 ∈ 𝐾 ↦ 𝐸) ∈ 𝑆)
17	4, 16	eqeltrd 2701	1 ⊢ (𝜑 → ∪ 𝑘 ∈ 𝐾 𝐸 ∈ 𝑆)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ⊆ wss 3574  𝒫 cpw 4158  ∪ cuni 4436  ∪ ciun 4520   class class class wbr 4653   ↦ cmpt 4729  ran crn 5115  ωcom 7065   ≼ cdom 7953  SAlgcsalg 40528

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

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-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-card 8765  df-acn 8768  df-salg 40529

This theorem is referenced by:  saliincl  40545  subsaliuncl  40576  meaiunlelem  40685  meaiuninclem  40697  meaiininclem  40700  caratheodory  40742  opnvonmbllem2  40847  ctvonmbl  40903  vonct  40907  smfaddlem2  40972  smflimlem1  40979  smfresal  40995  smfmullem4  41001