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Theorem bnj545 28927
Description: Technical lemma for bnj852 28953. 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 970 . . . . . . . . 9  |-  ( ta 
->  f  Fn  m
)
3 bnj545.5 . . . . . . . . . 10  |-  ( si  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  m )
)
43simp1bi 970 . . . . . . . . 9  |-  ( si  ->  m  e.  D )
52, 4anim12i 549 . . . . . . . 8  |-  ( ( ta  /\  si )  ->  ( f  Fn  m  /\  m  e.  D
) )
653adant1 973 . . . . . . 7  |-  ( ( R  FrSe  A  /\  ta  /\  si )  -> 
( f  Fn  m  /\  m  e.  D
) )
7 bnj545.2 . . . . . . . . 9  |-  D  =  ( om  \  { (/)
} )
87bnj529 28770 . . . . . . . 8  |-  ( m  e.  D  ->  (/)  e.  m
)
9 fndm 5343 . . . . . . . 8  |-  ( f  Fn  m  ->  dom  f  =  m )
10 eleq2 2344 . . . . . . . . 9  |-  ( dom  f  =  m  -> 
( (/)  e.  dom  f  <->  (/)  e.  m ) )
1110biimparc 473 . . . . . . . 8  |-  ( (
(/)  e.  m  /\  dom  f  =  m
)  ->  (/)  e.  dom  f )
128, 9, 11syl2anr 464 . . . . . . 7  |-  ( ( f  Fn  m  /\  m  e.  D )  -> 
(/)  e.  dom  f )
136, 12syl 15 . . . . . 6  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  (/) 
e.  dom  f )
14 bnj545.6 . . . . . . 7  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  G  Fn  n )
1514bnj930 28801 . . . . . 6  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  Fun  G )
1613, 15jca 518 . . . . 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 28802 . . . . 5  |-  f  C_  G
1916, 18jctil 523 . . . 4  |-  ( ( R  FrSe  A  /\  ta  /\  si )  -> 
( f  C_  G  /\  ( (/)  e.  dom  f  /\  Fun  G ) ) )
20 df-3an 936 . . . . 5  |-  ( (
(/)  e.  dom  f  /\  Fun  G  /\  f  C_  G )  <->  ( ( (/) 
e.  dom  f  /\  Fun  G )  /\  f  C_  G ) )
21 3anrot 939 . . . . 5  |-  ( (
(/)  e.  dom  f  /\  Fun  G  /\  f  C_  G )  <->  ( Fun  G  /\  f  C_  G  /\  (/)  e.  dom  f
) )
22 ancom 437 . . . . 5  |-  ( ( ( (/)  e.  dom  f  /\  Fun  G )  /\  f  C_  G
)  <->  ( f  C_  G  /\  ( (/)  e.  dom  f  /\  Fun  G ) ) )
2320, 21, 223bitr3i 266 . . . 4  |-  ( ( Fun  G  /\  f  C_  G  /\  (/)  e.  dom  f )  <->  ( f  C_  G  /\  ( (/)  e.  dom  f  /\  Fun  G ) ) )
2419, 23sylibr 203 . . 3  |-  ( ( R  FrSe  A  /\  ta  /\  si )  -> 
( Fun  G  /\  f  C_  G  /\  (/)  e.  dom  f ) )
25 funssfv 5543 . . 3  |-  ( ( Fun  G  /\  f  C_  G  /\  (/)  e.  dom  f )  ->  ( G `  (/) )  =  ( f `  (/) ) )
2624, 25syl 15 . 2  |-  ( ( R  FrSe  A  /\  ta  /\  si )  -> 
( G `  (/) )  =  ( f `  (/) ) )
271simp2bi 971 . . 3  |-  ( ta 
->  ph' )
28273ad2ant2 977 . 2  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  ph' )
29 bnj545.1 . . . 4  |-  ( ph'  <->  (
f `  (/) )  = 
pred ( x ,  A ,  R ) )
30 eqtr 2300 . . . 4  |-  ( ( ( G `  (/) )  =  ( f `  (/) )  /\  ( f `  (/) )  = 
pred ( x ,  A ,  R ) )  ->  ( G `  (/) )  =  pred ( x ,  A ,  R ) )
3129, 30sylan2b 461 . . 3  |-  ( ( ( G `  (/) )  =  ( f `  (/) )  /\  ph' )  ->  ( G `  (/) )  =  pred ( x ,  A ,  R ) )
32 bnj545.7 . . 3  |-  ( ph"  <->  ( G `  (/) )  =  pred ( x ,  A ,  R ) )
3331, 32sylibr 203 . 2  |-  ( ( ( G `  (/) )  =  ( f `  (/) )  /\  ph' )  ->  ph" )
3426, 28, 33syl2anc 642 1  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  ph" )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    \ cdif 3149    u. cun 3150    C_ wss 3152   (/)c0 3455   {csn 3640   <.cop 3643   U_ciun 3905   suc csuc 4394   omcom 4656   dom cdm 4689   Fun wfun 5249    Fn wfn 5250   ` cfv 5255    predc-bnj14 28713    FrSe w-bnj15 28717
This theorem is referenced by:  bnj600  28951  bnj908  28963
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-res 4701  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263
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