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Theorem bnj97 28660
Description: Technical lemma for bnj150 28670. 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.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj96.1  |-  F  =  { <. (/) ,  pred (
x ,  A ,  R ) >. }
Assertion
Ref Expression
bnj97  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  ( F `  (/) )  = 
pred ( x ,  A ,  R ) )
Distinct variable groups:    x, A    x, R
Allowed substitution hint:    F( x)

Proof of Theorem bnj97
StepHypRef Expression
1 bnj93 28657 . . 3  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  pred ( x ,  A ,  R )  e.  _V )
2 0ex 4231 . . . . 5  |-  (/)  e.  _V
32bnj519 28526 . . . 4  |-  (  pred ( x ,  A ,  R )  e.  _V  ->  Fun  { <. (/) ,  pred ( x ,  A ,  R ) >. } )
4 bnj96.1 . . . . 5  |-  F  =  { <. (/) ,  pred (
x ,  A ,  R ) >. }
54funeqi 5357 . . . 4  |-  ( Fun 
F  <->  Fun  { <. (/) ,  pred ( x ,  A ,  R ) >. } )
63, 5sylibr 203 . . 3  |-  (  pred ( x ,  A ,  R )  e.  _V  ->  Fun  F )
71, 6syl 15 . 2  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  Fun  F )
8 opex 4319 . . . 4  |-  <. (/) ,  pred ( x ,  A ,  R ) >.  e.  _V
98snid 3743 . . 3  |-  <. (/) ,  pred ( x ,  A ,  R ) >.  e.  { <.
(/) ,  pred ( x ,  A ,  R
) >. }
109, 4eleqtrri 2431 . 2  |-  <. (/) ,  pred ( x ,  A ,  R ) >.  e.  F
11 funopfv 5645 . 2  |-  ( Fun 
F  ->  ( <. (/)
,  pred ( x ,  A ,  R )
>.  e.  F  ->  ( F `  (/) )  = 
pred ( x ,  A ,  R ) ) )
127, 10, 11ee10 1376 1  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  ( F `  (/) )  = 
pred ( x ,  A ,  R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710   _Vcvv 2864   (/)c0 3531   {csn 3716   <.cop 3719   Fun wfun 5331   ` cfv 5337    predc-bnj14 28475    FrSe w-bnj15 28479
This theorem is referenced by:  bnj150  28670
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-iota 5301  df-fun 5339  df-fv 5345  df-bnj13 28478  df-bnj15 28480
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