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Theorem bnj97 33359
Description: Technical lemma for bnj150 33369. 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 33356 . . 3  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  pred ( x ,  A ,  R )  e.  _V )
2 0ex 4583 . . . . 5  |-  (/)  e.  _V
32bnj519 33227 . . . 4  |-  (  pred ( x ,  A ,  R )  e.  _V  ->  Fun  { <. (/) ,  pred ( x ,  A ,  R ) >. } )
4 bnj96.1 . . . . 5  |-  F  =  { <. (/) ,  pred (
x ,  A ,  R ) >. }
54funeqi 5614 . . . 4  |-  ( Fun 
F  <->  Fun  { <. (/) ,  pred ( x ,  A ,  R ) >. } )
63, 5sylibr 212 . . 3  |-  (  pred ( x ,  A ,  R )  e.  _V  ->  Fun  F )
71, 6syl 16 . 2  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  Fun  F )
8 opex 4717 . . . 4  |-  <. (/) ,  pred ( x ,  A ,  R ) >.  e.  _V
98snid 4061 . . 3  |-  <. (/) ,  pred ( x ,  A ,  R ) >.  e.  { <.
(/) ,  pred ( x ,  A ,  R
) >. }
109, 4eleqtrri 2554 . 2  |-  <. (/) ,  pred ( x ,  A ,  R ) >.  e.  F
11 funopfv 5913 . 2  |-  ( Fun 
F  ->  ( <. (/)
,  pred ( x ,  A ,  R )
>.  e.  F  ->  ( F `  (/) )  = 
pred ( x ,  A ,  R ) ) )
127, 10, 11mpisyl 18 1  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  ( F `  (/) )  = 
pred ( x ,  A ,  R ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3118   (/)c0 3790   {csn 4033   <.cop 4039   Fun wfun 5588   ` cfv 5594    predc-bnj14 33176    FrSe w-bnj15 33180
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-iota 5557  df-fun 5596  df-fv 5602  df-bnj13 33179  df-bnj15 33181
This theorem is referenced by:  bnj150  33369
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