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Theorem dprddomcld 20021
Description: If a family of subgroups is a family of subgroups for an internal direct product, then it is indexed by a set. (Contributed by AV, 13-Jul-2019.)
Hypotheses
Ref Expression
dprddomcld.1 (𝜑𝐺dom DProd 𝑆)
dprddomcld.2 (𝜑 → dom 𝑆 = 𝐼)
Assertion
Ref Expression
dprddomcld (𝜑𝐼 ∈ V)

Proof of Theorem dprddomcld
StepHypRef Expression
1 dprddomcld.2 . 2 (𝜑 → dom 𝑆 = 𝐼)
2 dprddomcld.1 . 2 (𝜑𝐺dom DProd 𝑆)
3 df-nel 3047 . . . . 5 (dom 𝑆 ∉ V ↔ ¬ dom 𝑆 ∈ V)
4 dprddomprc 20020 . . . . 5 (dom 𝑆 ∉ V → ¬ 𝐺dom DProd 𝑆)
53, 4sylbir 235 . . . 4 (¬ dom 𝑆 ∈ V → ¬ 𝐺dom DProd 𝑆)
65con4i 114 . . 3 (𝐺dom DProd 𝑆 → dom 𝑆 ∈ V)
7 eleq1 2829 . . 3 (dom 𝑆 = 𝐼 → (dom 𝑆 ∈ V ↔ 𝐼 ∈ V))
86, 7imbitrid 244 . 2 (dom 𝑆 = 𝐼 → (𝐺dom DProd 𝑆𝐼 ∈ V))
91, 2, 8sylc 65 1 (𝜑𝐼 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1540  wcel 2108  wnel 3046  Vcvv 3480   class class class wbr 5143  dom cdm 5685   DProd cdprd 20013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696  df-oprab 7435  df-mpo 7436  df-dprd 20015
This theorem is referenced by:  dprdcntz  20028  dprddisj  20029  dprdw  20030  dprdwd  20031  dprdfid  20037  dprdfinv  20039  dprdfadd  20040  dprdfsub  20041  dprdfeq0  20042  dprdf11  20043  dprdlub  20046  dprdres  20048  dprdss  20049  dprdf1o  20052  dmdprdsplitlem  20057  dprddisj2  20059  dmdprdsplit2  20066  dpjfval  20075  dpjidcl  20078
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