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Theorem bnj545 33249
Description: Technical lemma for bnj852 33275. 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.)
Hypotheses
Ref Expression
bnj545.1  |-  ( ph'  <->  (
f `  (/) )  = 
pred ( x ,  A ,  R ) )
bnj545.2  |-  D  =  ( om  \  { (/)
} )
bnj545.3  |-  G  =  ( f  u.  { <. m ,  U_ y  e.  ( f `  p
)  pred ( y ,  A ,  R )
>. } )
bnj545.4  |-  ( ta  <->  ( f  Fn  m  /\  ph' 
/\  ps' ) )
bnj545.5  |-  ( si  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  m )
)
bnj545.6  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  G  Fn  n )
bnj545.7  |-  ( ph"  <->  ( G `  (/) )  =  pred ( x ,  A ,  R ) )
Assertion
Ref Expression
bnj545  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  ph" )

Proof of Theorem bnj545
StepHypRef Expression
1 bnj545.4 . . . . . . . . . 10  |-  ( ta  <->  ( f  Fn  m  /\  ph' 
/\  ps' ) )
21simp1bi 1011 . . . . . . . . 9  |-  ( ta 
->  f  Fn  m
)
3 bnj545.5 . . . . . . . . . 10  |-  ( si  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  m )
)
43simp1bi 1011 . . . . . . . . 9  |-  ( si  ->  m  e.  D )
52, 4anim12i 566 . . . . . . . 8  |-  ( ( ta  /\  si )  ->  ( f  Fn  m  /\  m  e.  D
) )
653adant1 1014 . . . . . . 7  |-  ( ( R  FrSe  A  /\  ta  /\  si )  -> 
( f  Fn  m  /\  m  e.  D
) )
7 bnj545.2 . . . . . . . . 9  |-  D  =  ( om  \  { (/)
} )
87bnj529 33094 . . . . . . . 8  |-  ( m  e.  D  ->  (/)  e.  m
)
9 fndm 5680 . . . . . . . 8  |-  ( f  Fn  m  ->  dom  f  =  m )
10 eleq2 2540 . . . . . . . . 9  |-  ( dom  f  =  m  -> 
( (/)  e.  dom  f  <->  (/)  e.  m ) )
1110biimparc 487 . . . . . . . 8  |-  ( (
(/)  e.  m  /\  dom  f  =  m
)  ->  (/)  e.  dom  f )
128, 9, 11syl2anr 478 . . . . . . 7  |-  ( ( f  Fn  m  /\  m  e.  D )  -> 
(/)  e.  dom  f )
136, 12syl 16 . . . . . 6  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  (/) 
e.  dom  f )
14 bnj545.6 . . . . . . 7  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  G  Fn  n )
1514bnj930 33124 . . . . . 6  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  Fun  G )
1613, 15jca 532 . . . . 5  |-  ( ( R  FrSe  A  /\  ta  /\  si )  -> 
( (/)  e.  dom  f  /\  Fun  G ) )
17 bnj545.3 . . . . . 6  |-  G  =  ( f  u.  { <. m ,  U_ y  e.  ( f `  p
)  pred ( y ,  A ,  R )
>. } )
1817bnj931 33125 . . . . 5  |-  f  C_  G
1916, 18jctil 537 . . . 4  |-  ( ( R  FrSe  A  /\  ta  /\  si )  -> 
( f  C_  G  /\  ( (/)  e.  dom  f  /\  Fun  G ) ) )
20 df-3an 975 . . . . 5  |-  ( (
(/)  e.  dom  f  /\  Fun  G  /\  f  C_  G )  <->  ( ( (/) 
e.  dom  f  /\  Fun  G )  /\  f  C_  G ) )
21 3anrot 978 . . . . 5  |-  ( (
(/)  e.  dom  f  /\  Fun  G  /\  f  C_  G )  <->  ( Fun  G  /\  f  C_  G  /\  (/)  e.  dom  f
) )
22 ancom 450 . . . . 5  |-  ( ( ( (/)  e.  dom  f  /\  Fun  G )  /\  f  C_  G
)  <->  ( f  C_  G  /\  ( (/)  e.  dom  f  /\  Fun  G ) ) )
2320, 21, 223bitr3i 275 . . . 4  |-  ( ( Fun  G  /\  f  C_  G  /\  (/)  e.  dom  f )  <->  ( f  C_  G  /\  ( (/)  e.  dom  f  /\  Fun  G ) ) )
2419, 23sylibr 212 . . 3  |-  ( ( R  FrSe  A  /\  ta  /\  si )  -> 
( Fun  G  /\  f  C_  G  /\  (/)  e.  dom  f ) )
25 funssfv 5881 . . 3  |-  ( ( Fun  G  /\  f  C_  G  /\  (/)  e.  dom  f )  ->  ( G `  (/) )  =  ( f `  (/) ) )
2624, 25syl 16 . 2  |-  ( ( R  FrSe  A  /\  ta  /\  si )  -> 
( G `  (/) )  =  ( f `  (/) ) )
271simp2bi 1012 . . 3  |-  ( ta 
->  ph' )
28273ad2ant2 1018 . 2  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  ph' )
29 bnj545.1 . . . 4  |-  ( ph'  <->  (
f `  (/) )  = 
pred ( x ,  A ,  R ) )
30 eqtr 2493 . . . 4  |-  ( ( ( G `  (/) )  =  ( f `  (/) )  /\  ( f `  (/) )  = 
pred ( x ,  A ,  R ) )  ->  ( G `  (/) )  =  pred ( x ,  A ,  R ) )
3129, 30sylan2b 475 . . 3  |-  ( ( ( G `  (/) )  =  ( f `  (/) )  /\  ph' )  ->  ( G `  (/) )  =  pred ( x ,  A ,  R ) )
32 bnj545.7 . . 3  |-  ( ph"  <->  ( G `  (/) )  =  pred ( x ,  A ,  R ) )
3331, 32sylibr 212 . 2  |-  ( ( ( G `  (/) )  =  ( f `  (/) )  /\  ph' )  ->  ph" )
3426, 28, 33syl2anc 661 1  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  ph" )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    \ cdif 3473    u. cun 3474    C_ wss 3476   (/)c0 3785   {csn 4027   <.cop 4033   U_ciun 4325   suc csuc 4880   dom cdm 4999   Fun wfun 5582    Fn wfn 5583   ` cfv 5588   omcom 6685    predc-bnj14 33037    FrSe w-bnj15 33041
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-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  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 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-res 5011  df-iota 5551  df-fun 5590  df-fn 5591  df-fv 5596  df-om 6686
This theorem is referenced by:  bnj600  33273  bnj908  33285
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