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Theorem dprddomcld 19871
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 3048 . . . . 5 (dom 𝑆 ∉ V ↔ ¬ dom 𝑆 ∈ V)
4 dprddomprc 19870 . . . . 5 (dom 𝑆 ∉ V → ¬ 𝐺dom DProd 𝑆)
53, 4sylbir 234 . . . 4 (¬ dom 𝑆 ∈ V → ¬ 𝐺dom DProd 𝑆)
65con4i 114 . . 3 (𝐺dom DProd 𝑆 → dom 𝑆 ∈ V)
7 eleq1 2822 . . 3 (dom 𝑆 = 𝐼 → (dom 𝑆 ∈ V ↔ 𝐼 ∈ V))
86, 7imbitrid 243 . 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 1542  wcel 2107  wnel 3047  Vcvv 3475   class class class wbr 5149  dom cdm 5677   DProd cdprd 19863
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-nel 3048  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-cnv 5685  df-dm 5687  df-rn 5688  df-oprab 7413  df-mpo 7414  df-dprd 19865
This theorem is referenced by:  dprdcntz  19878  dprddisj  19879  dprdw  19880  dprdwd  19881  dprdfid  19887  dprdfinv  19889  dprdfadd  19890  dprdfsub  19891  dprdfeq0  19892  dprdf11  19893  dprdlub  19896  dprdres  19898  dprdss  19899  dprdf1o  19902  dmdprdsplitlem  19907  dprddisj2  19909  dmdprdsplit2  19916  dpjfval  19925  dpjidcl  19928
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