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Theorem elpwiuncl 29331
Description: Closure of indexed union with regard to elementhood to a power set. (Contributed by Thierry Arnoux, 27-May-2020.)
Hypotheses
Ref Expression
elpwiuncl.1 (𝜑𝐴𝑉)
elpwiuncl.2 ((𝜑𝑘𝐴) → 𝐵 ∈ 𝒫 𝐶)
Assertion
Ref Expression
elpwiuncl (𝜑 𝑘𝐴 𝐵 ∈ 𝒫 𝐶)
Distinct variable groups:   𝐴,𝑘   𝐶,𝑘   𝜑,𝑘
Allowed substitution hints:   𝐵(𝑘)   𝑉(𝑘)

Proof of Theorem elpwiuncl
StepHypRef Expression
1 elpwiuncl.2 . . . . 5 ((𝜑𝑘𝐴) → 𝐵 ∈ 𝒫 𝐶)
21elpwid 4161 . . . 4 ((𝜑𝑘𝐴) → 𝐵𝐶)
32ralrimiva 2963 . . 3 (𝜑 → ∀𝑘𝐴 𝐵𝐶)
4 iunss 4552 . . 3 ( 𝑘𝐴 𝐵𝐶 ↔ ∀𝑘𝐴 𝐵𝐶)
53, 4sylibr 224 . 2 (𝜑 𝑘𝐴 𝐵𝐶)
6 elpwiuncl.1 . . . 4 (𝜑𝐴𝑉)
71ralrimiva 2963 . . . 4 (𝜑 → ∀𝑘𝐴 𝐵 ∈ 𝒫 𝐶)
86, 7jca 554 . . 3 (𝜑 → (𝐴𝑉 ∧ ∀𝑘𝐴 𝐵 ∈ 𝒫 𝐶))
9 iunexg 7128 . . 3 ((𝐴𝑉 ∧ ∀𝑘𝐴 𝐵 ∈ 𝒫 𝐶) → 𝑘𝐴 𝐵 ∈ V)
10 elpwg 4157 . . 3 ( 𝑘𝐴 𝐵 ∈ V → ( 𝑘𝐴 𝐵 ∈ 𝒫 𝐶 𝑘𝐴 𝐵𝐶))
118, 9, 103syl 18 . 2 (𝜑 → ( 𝑘𝐴 𝐵 ∈ 𝒫 𝐶 𝑘𝐴 𝐵𝐶))
125, 11mpbird 247 1 (𝜑 𝑘𝐴 𝐵 ∈ 𝒫 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wcel 1988  wral 2909  Vcvv 3195  wss 3567  𝒫 cpw 4149   ciun 4511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884
This theorem is referenced by:  carsggect  30354  carsgclctunlem2  30355
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