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Theorem ismri2 13534
Description: Criterion for a subset of the base set in a Moore system to be independent. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
ismri2.1  |-  N  =  (mrCls `  A )
ismri2.2  |-  I  =  (mrInd `  A )
Assertion
Ref Expression
ismri2  |-  ( ( A  e.  (Moore `  X )  /\  S  C_  X )  ->  ( S  e.  I  <->  A. x  e.  S  -.  x  e.  ( N `  ( S  \  { x }
) ) ) )
Distinct variable groups:    x, A    x, S
Allowed substitution hints:    I( x)    N( x)    X( x)

Proof of Theorem ismri2
StepHypRef Expression
1 ismri2.1 . . 3  |-  N  =  (mrCls `  A )
2 ismri2.2 . . 3  |-  I  =  (mrInd `  A )
31, 2ismri 13533 . 2  |-  ( A  e.  (Moore `  X
)  ->  ( S  e.  I  <->  ( S  C_  X  /\  A. x  e.  S  -.  x  e.  ( N `  ( S  \  { x }
) ) ) ) )
43baibd 875 1  |-  ( ( A  e.  (Moore `  X )  /\  S  C_  X )  ->  ( S  e.  I  <->  A. x  e.  S  -.  x  e.  ( N `  ( S  \  { x }
) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    \ cdif 3149    C_ wss 3152   {csn 3640   ` cfv 5255  Moorecmre 13484  mrClscmrc 13485  mrIndcmri 13486
This theorem is referenced by:  ismri2d  13535
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fv 5263  df-mre 13488  df-mri 13490
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