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Theorem dprddomcld 20036
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 3045 . . . . 5 (dom 𝑆 ∉ V ↔ ¬ dom 𝑆 ∈ V)
4 dprddomprc 20035 . . . . 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 2827 . . 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 1537  wcel 2106  wnel 3044  Vcvv 3478   class class class wbr 5148  dom cdm 5689   DProd cdprd 20028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-oprab 7435  df-mpo 7436  df-dprd 20030
This theorem is referenced by:  dprdcntz  20043  dprddisj  20044  dprdw  20045  dprdwd  20046  dprdfid  20052  dprdfinv  20054  dprdfadd  20055  dprdfsub  20056  dprdfeq0  20057  dprdf11  20058  dprdlub  20061  dprdres  20063  dprdss  20064  dprdf1o  20067  dmdprdsplitlem  20072  dprddisj2  20074  dmdprdsplit2  20081  dpjfval  20090  dpjidcl  20093
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