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Theorem bnj96 29464
Description: Technical lemma for bnj150 29475. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Revised by Mario Carneiro, 6-May-2015.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj96.1  |-  F  =  { <. (/) ,  pred (
x ,  A ,  R ) >. }
Assertion
Ref Expression
bnj96  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  dom  F  =  1o )
Distinct variable groups:    x, A    x, R
Allowed substitution hint:    F( x)

Proof of Theorem bnj96
StepHypRef Expression
1 bnj93 29462 . . 3  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  pred ( x ,  A ,  R )  e.  _V )
2 dmsnopg 5327 . . 3  |-  (  pred ( x ,  A ,  R )  e.  _V  ->  dom  { <. (/) ,  pred ( x ,  A ,  R ) >. }  =  { (/) } )
31, 2syl 17 . 2  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  dom  { <. (/) ,  pred ( x ,  A ,  R ) >. }  =  { (/) } )
4 bnj96.1 . . 3  |-  F  =  { <. (/) ,  pred (
x ,  A ,  R ) >. }
54dmeqi 5056 . 2  |-  dom  F  =  dom  { <. (/) ,  pred ( x ,  A ,  R ) >. }
6 df1o2 7202 . 2  |-  1o  =  { (/) }
73, 5, 63eqtr4g 2495 1  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  dom  F  =  1o )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   _Vcvv 3087   (/)c0 3767   {csn 4002   <.cop 4008   dom cdm 4854   1oc1o 7183    predc-bnj14 29281    FrSe w-bnj15 29285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-br 4427  df-dm 4864  df-suc 5448  df-1o 7190  df-bnj13 29284  df-bnj15 29286
This theorem is referenced by:  bnj150  29475
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