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Theorem bnj97 28955
Description: Technical lemma for bnj150 28965. 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 28952 . . 3  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  pred ( x ,  A ,  R )  e.  _V )
2 0ex 4307 . . . . 5  |-  (/)  e.  _V
32bnj519 28821 . . . 4  |-  (  pred ( x ,  A ,  R )  e.  _V  ->  Fun  { <. (/) ,  pred ( x ,  A ,  R ) >. } )
4 bnj96.1 . . . . 5  |-  F  =  { <. (/) ,  pred (
x ,  A ,  R ) >. }
54funeqi 5441 . . . 4  |-  ( Fun 
F  <->  Fun  { <. (/) ,  pred ( x ,  A ,  R ) >. } )
63, 5sylibr 204 . . 3  |-  (  pred ( x ,  A ,  R )  e.  _V  ->  Fun  F )
71, 6syl 16 . 2  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  Fun  F )
8 opex 4395 . . . 4  |-  <. (/) ,  pred ( x ,  A ,  R ) >.  e.  _V
98snid 3809 . . 3  |-  <. (/) ,  pred ( x ,  A ,  R ) >.  e.  { <.
(/) ,  pred ( x ,  A ,  R
) >. }
109, 4eleqtrri 2485 . 2  |-  <. (/) ,  pred ( x ,  A ,  R ) >.  e.  F
11 funopfv 5733 . 2  |-  ( Fun 
F  ->  ( <. (/)
,  pred ( x ,  A ,  R )
>.  e.  F  ->  ( F `  (/) )  = 
pred ( x ,  A ,  R ) ) )
127, 10, 11ee10 1382 1  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  ( F `  (/) )  = 
pred ( x ,  A ,  R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2924   (/)c0 3596   {csn 3782   <.cop 3785   Fun wfun 5415   ` cfv 5421    predc-bnj14 28770    FrSe w-bnj15 28774
This theorem is referenced by:  bnj150  28965
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5385  df-fun 5423  df-fv 5429  df-bnj13 28773  df-bnj15 28775
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