Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  dprdval0prc Structured version   Visualization version   GIF version

Theorem dprdval0prc 19115
 Description: The internal direct product of a family of subgroups indexed by a proper class is empty. (Contributed by AV, 13-Jul-2019.)
Assertion
Ref Expression
dprdval0prc (dom 𝑆 ∉ V → (𝐺 DProd 𝑆) = ∅)

Proof of Theorem dprdval0prc
StepHypRef Expression
1 df-nel 3116 . . 3 (dom 𝑆 ∉ V ↔ ¬ dom 𝑆 ∈ V)
2 dmexg 7599 . . . 4 (𝑆 ∈ V → dom 𝑆 ∈ V)
32con3i 157 . . 3 (¬ dom 𝑆 ∈ V → ¬ 𝑆 ∈ V)
41, 3sylbi 220 . 2 (dom 𝑆 ∉ V → ¬ 𝑆 ∈ V)
5 reldmdprd 19110 . . 3 Rel dom DProd
65ovprc2 7180 . 2 𝑆 ∈ V → (𝐺 DProd 𝑆) = ∅)
74, 6syl 17 1 (dom 𝑆 ∉ V → (𝐺 DProd 𝑆) = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1538   ∈ wcel 2114   ∉ wnel 3115  Vcvv 3469  ∅c0 4265  dom cdm 5532  (class class class)co 7140   DProd cdprd 19106 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-nel 3116  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-xp 5538  df-rel 5539  df-cnv 5540  df-dm 5542  df-rn 5543  df-iota 6293  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-dprd 19108 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator