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Theorem bnj545 29340
Description: Technical lemma for bnj852 29366. 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 973 . . . . . . . . 9  |-  ( ta 
->  f  Fn  m
)
3 bnj545.5 . . . . . . . . . 10  |-  ( si  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  m )
)
43simp1bi 973 . . . . . . . . 9  |-  ( si  ->  m  e.  D )
52, 4anim12i 551 . . . . . . . 8  |-  ( ( ta  /\  si )  ->  ( f  Fn  m  /\  m  e.  D
) )
653adant1 976 . . . . . . 7  |-  ( ( R  FrSe  A  /\  ta  /\  si )  -> 
( f  Fn  m  /\  m  e.  D
) )
7 bnj545.2 . . . . . . . . 9  |-  D  =  ( om  \  { (/)
} )
87bnj529 29183 . . . . . . . 8  |-  ( m  e.  D  ->  (/)  e.  m
)
9 fndm 5547 . . . . . . . 8  |-  ( f  Fn  m  ->  dom  f  =  m )
10 eleq2 2499 . . . . . . . . 9  |-  ( dom  f  =  m  -> 
( (/)  e.  dom  f  <->  (/)  e.  m ) )
1110biimparc 475 . . . . . . . 8  |-  ( (
(/)  e.  m  /\  dom  f  =  m
)  ->  (/)  e.  dom  f )
128, 9, 11syl2anr 466 . . . . . . 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 29214 . . . . . 6  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  Fun  G )
1613, 15jca 520 . . . . 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 29215 . . . . 5  |-  f  C_  G
1916, 18jctil 525 . . . 4  |-  ( ( R  FrSe  A  /\  ta  /\  si )  -> 
( f  C_  G  /\  ( (/)  e.  dom  f  /\  Fun  G ) ) )
20 df-3an 939 . . . . 5  |-  ( (
(/)  e.  dom  f  /\  Fun  G  /\  f  C_  G )  <->  ( ( (/) 
e.  dom  f  /\  Fun  G )  /\  f  C_  G ) )
21 3anrot 942 . . . . 5  |-  ( (
(/)  e.  dom  f  /\  Fun  G  /\  f  C_  G )  <->  ( Fun  G  /\  f  C_  G  /\  (/)  e.  dom  f
) )
22 ancom 439 . . . . 5  |-  ( ( ( (/)  e.  dom  f  /\  Fun  G )  /\  f  C_  G
)  <->  ( f  C_  G  /\  ( (/)  e.  dom  f  /\  Fun  G ) ) )
2320, 21, 223bitr3i 268 . . . 4  |-  ( ( Fun  G  /\  f  C_  G  /\  (/)  e.  dom  f )  <->  ( f  C_  G  /\  ( (/)  e.  dom  f  /\  Fun  G ) ) )
2419, 23sylibr 205 . . 3  |-  ( ( R  FrSe  A  /\  ta  /\  si )  -> 
( Fun  G  /\  f  C_  G  /\  (/)  e.  dom  f ) )
25 funssfv 5749 . . 3  |-  ( ( Fun  G  /\  f  C_  G  /\  (/)  e.  dom  f )  ->  ( G `  (/) )  =  ( f `  (/) ) )
2624, 25syl 16 . 2  |-  ( ( R  FrSe  A  /\  ta  /\  si )  -> 
( G `  (/) )  =  ( f `  (/) ) )
271simp2bi 974 . . 3  |-  ( ta 
->  ph' )
28273ad2ant2 980 . 2  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  ph' )
29 bnj545.1 . . . 4  |-  ( ph'  <->  (
f `  (/) )  = 
pred ( x ,  A ,  R ) )
30 eqtr 2455 . . . 4  |-  ( ( ( G `  (/) )  =  ( f `  (/) )  /\  ( f `  (/) )  = 
pred ( x ,  A ,  R ) )  ->  ( G `  (/) )  =  pred ( x ,  A ,  R ) )
3129, 30sylan2b 463 . . 3  |-  ( ( ( G `  (/) )  =  ( f `  (/) )  /\  ph' )  ->  ( G `  (/) )  =  pred ( x ,  A ,  R ) )
32 bnj545.7 . . 3  |-  ( ph"  <->  ( G `  (/) )  =  pred ( x ,  A ,  R ) )
3331, 32sylibr 205 . 2  |-  ( ( ( G `  (/) )  =  ( f `  (/) )  /\  ph' )  ->  ph" )
3426, 28, 33syl2anc 644 1  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  ph" )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    \ cdif 3319    u. cun 3320    C_ wss 3322   (/)c0 3630   {csn 3816   <.cop 3819   U_ciun 4095   suc csuc 4586   omcom 4848   dom cdm 4881   Fun wfun 5451    Fn wfn 5452   ` cfv 5457    predc-bnj14 29126    FrSe w-bnj15 29130
This theorem is referenced by:  bnj600  29364  bnj908  29376
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-res 4893  df-iota 5421  df-fun 5459  df-fn 5460  df-fv 5465
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