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Theorem bnj97 29473
Description: Technical lemma for bnj150 29483. 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 29470 . . 3  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  pred ( x ,  A ,  R )  e.  _V )
2 0ex 4557 . . . . 5  |-  (/)  e.  _V
32bnj519 29340 . . . 4  |-  (  pred ( x ,  A ,  R )  e.  _V  ->  Fun  { <. (/) ,  pred ( x ,  A ,  R ) >. } )
4 bnj96.1 . . . . 5  |-  F  =  { <. (/) ,  pred (
x ,  A ,  R ) >. }
54funeqi 5621 . . . 4  |-  ( Fun 
F  <->  Fun  { <. (/) ,  pred ( x ,  A ,  R ) >. } )
63, 5sylibr 215 . . 3  |-  (  pred ( x ,  A ,  R )  e.  _V  ->  Fun  F )
71, 6syl 17 . 2  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  Fun  F )
8 opex 4686 . . . 4  |-  <. (/) ,  pred ( x ,  A ,  R ) >.  e.  _V
98snid 4030 . . 3  |-  <. (/) ,  pred ( x ,  A ,  R ) >.  e.  { <.
(/) ,  pred ( x ,  A ,  R
) >. }
109, 4eleqtrri 2516 . 2  |-  <. (/) ,  pred ( x ,  A ,  R ) >.  e.  F
11 funopfv 5920 . 2  |-  ( Fun 
F  ->  ( <. (/)
,  pred ( x ,  A ,  R )
>.  e.  F  ->  ( F `  (/) )  = 
pred ( x ,  A ,  R ) ) )
127, 10, 11mpisyl 22 1  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  ( F `  (/) )  = 
pred ( x ,  A ,  R ) )
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   Fun wfun 5595   ` cfv 5601    predc-bnj14 29289    FrSe w-bnj15 29293
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-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  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-uni 4223  df-br 4427  df-opab 4485  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-iota 5565  df-fun 5603  df-fv 5609  df-bnj13 29292  df-bnj15 29294
This theorem is referenced by:  bnj150  29483
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