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Theorem dprddomcld 16590
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 2647 . . . . 5  |-  ( dom 
S  e/  _V  <->  -.  dom  S  e.  _V )
4 dprddomprc 16589 . . . . 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 2523 . . 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 1370    e. wcel 1758    e/ wnel 2645   _Vcvv 3070   class class class wbr 4392   dom cdm 4940   DProd cdprd 16582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-xp 4946  df-rel 4947  df-cnv 4948  df-dm 4950  df-rn 4951  df-oprab 6196  df-mpt2 6197  df-dprd 16584
This theorem is referenced by:  dprdfid  16614  dprdfinv  16616  dprdfadd  16617  dprdfsub  16618  dprdlub  16630  dprdss  16633  dmdprdsplitlem  16641  dprddisj2  16644  dpjidcl  16664
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