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Theorem domintreflemb 25062
Description: In a reflexive class  R, an element  A belongs to the field iff the pair  <. A ,  A >. belongs to  R. (Contributed by FL, 30-Dec-2011.)
Assertion
Ref Expression
domintreflemb  |-  ( ( A  e.  B  /\  A. x  e.  dom  R  x R x )  -> 
( A  e.  dom  R  <-> 
A R A ) )
Distinct variable groups:    x, A    x, R
Allowed substitution hint:    B( x)

Proof of Theorem domintreflemb
StepHypRef Expression
1 id 19 . . . . 5  |-  ( x  =  A  ->  x  =  A )
21, 1breq12d 4036 . . . 4  |-  ( x  =  A  ->  (
x R x  <->  A R A ) )
32rspccv 2881 . . 3  |-  ( A. x  e.  dom  R  x R x  ->  ( A  e.  dom  R  ->  A R A ) )
43adantl 452 . 2  |-  ( ( A  e.  B  /\  A. x  e.  dom  R  x R x )  -> 
( A  e.  dom  R  ->  A R A ) )
5 breldmg 4884 . . . . 5  |-  ( ( A  e.  B  /\  A  e.  B  /\  A R A )  ->  A  e.  dom  R )
653expia 1153 . . . 4  |-  ( ( A  e.  B  /\  A  e.  B )  ->  ( A R A  ->  A  e.  dom  R ) )
76anidms 626 . . 3  |-  ( A  e.  B  ->  ( A R A  ->  A  e.  dom  R ) )
87adantr 451 . 2  |-  ( ( A  e.  B  /\  A. x  e.  dom  R  x R x )  -> 
( A R A  ->  A  e.  dom  R ) )
94, 8impbid 183 1  |-  ( ( A  e.  B  /\  A. x  e.  dom  R  x R x )  -> 
( A  e.  dom  R  <-> 
A R A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   class class class wbr 4023   dom cdm 4689
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
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-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-dm 4699
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