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Theorem bnj97 31864
Description: Technical lemma for bnj150 31874. 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 31861 . . 3  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  pred ( x ,  A ,  R )  e.  _V )
2 0ex 4427 . . . . 5  |-  (/)  e.  _V
32bnj519 31732 . . . 4  |-  (  pred ( x ,  A ,  R )  e.  _V  ->  Fun  { <. (/) ,  pred ( x ,  A ,  R ) >. } )
4 bnj96.1 . . . . 5  |-  F  =  { <. (/) ,  pred (
x ,  A ,  R ) >. }
54funeqi 5443 . . . 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 4561 . . . 4  |-  <. (/) ,  pred ( x ,  A ,  R ) >.  e.  _V
98snid 3910 . . 3  |-  <. (/) ,  pred ( x ,  A ,  R ) >.  e.  { <.
(/) ,  pred ( x ,  A ,  R
) >. }
109, 4eleqtrri 2516 . 2  |-  <. (/) ,  pred ( x ,  A ,  R ) >.  e.  F
11 funopfv 5736 . 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 1369    e. wcel 1756   _Vcvv 2977   (/)c0 3642   {csn 3882   <.cop 3888   Fun wfun 5417   ` cfv 5423    predc-bnj14 31681    FrSe w-bnj15 31685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5386  df-fun 5425  df-fv 5431  df-bnj13 31684  df-bnj15 31686
This theorem is referenced by:  bnj150  31874
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