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Theorem dprddomcld 16816
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  |-  ( ph  ->  G dom DProd  S )
dprddomcld.2  |-  ( ph  ->  dom  S  =  I )
Assertion
Ref Expression
dprddomcld  |-  ( ph  ->  I  e.  _V )

Proof of Theorem dprddomcld
StepHypRef Expression
1 dprddomcld.2 . 2  |-  ( ph  ->  dom  S  =  I )
2 dprddomcld.1 . 2  |-  ( ph  ->  G dom DProd  S )
3 df-nel 2658 . . . . 5  |-  ( dom 
S  e/  _V  <->  -.  dom  S  e.  _V )
4 dprddomprc 16815 . . . . 5  |-  ( dom 
S  e/  _V  ->  -.  G dom DProd  S )
53, 4sylbir 213 . . . 4  |-  ( -. 
dom  S  e.  _V  ->  -.  G dom DProd  S )
65con4i 130 . . 3  |-  ( G dom DProd  S  ->  dom  S  e.  _V )
7 eleq1 2532 . . 3  |-  ( dom 
S  =  I  -> 
( dom  S  e.  _V 
<->  I  e.  _V )
)
86, 7syl5ib 219 . 2  |-  ( dom 
S  =  I  -> 
( G dom DProd  S  ->  I  e.  _V )
)
91, 2, 8sylc 60 1  |-  ( ph  ->  I  e.  _V )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1374    e. wcel 1762    e/ wnel 2656   _Vcvv 3106   class class class wbr 4440   dom cdm 4992   DProd cdprd 16808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-xp 4998  df-rel 4999  df-cnv 5000  df-dm 5002  df-rn 5003  df-oprab 6279  df-mpt2 6280  df-dprd 16810
This theorem is referenced by:  dprdfid  16840  dprdfinv  16842  dprdfadd  16843  dprdfsub  16844  dprdlub  16856  dprdss  16859  dmdprdsplitlem  16867  dprddisj2  16870  dpjidcl  16890
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