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Theorem bnj545 29193
Description: Technical lemma for bnj852 29219. 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 972 . . . . . . . . 9  |-  ( ta 
->  f  Fn  m
)
3 bnj545.5 . . . . . . . . . 10  |-  ( si  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  m )
)
43simp1bi 972 . . . . . . . . 9  |-  ( si  ->  m  e.  D )
52, 4anim12i 550 . . . . . . . 8  |-  ( ( ta  /\  si )  ->  ( f  Fn  m  /\  m  e.  D
) )
653adant1 975 . . . . . . 7  |-  ( ( R  FrSe  A  /\  ta  /\  si )  -> 
( f  Fn  m  /\  m  e.  D
) )
7 bnj545.2 . . . . . . . . 9  |-  D  =  ( om  \  { (/)
} )
87bnj529 29036 . . . . . . . 8  |-  ( m  e.  D  ->  (/)  e.  m
)
9 fndm 5536 . . . . . . . 8  |-  ( f  Fn  m  ->  dom  f  =  m )
10 eleq2 2496 . . . . . . . . 9  |-  ( dom  f  =  m  -> 
( (/)  e.  dom  f  <->  (/)  e.  m ) )
1110biimparc 474 . . . . . . . 8  |-  ( (
(/)  e.  m  /\  dom  f  =  m
)  ->  (/)  e.  dom  f )
128, 9, 11syl2anr 465 . . . . . . 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 29067 . . . . . 6  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  Fun  G )
1613, 15jca 519 . . . . 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 29068 . . . . 5  |-  f  C_  G
1916, 18jctil 524 . . . 4  |-  ( ( R  FrSe  A  /\  ta  /\  si )  -> 
( f  C_  G  /\  ( (/)  e.  dom  f  /\  Fun  G ) ) )
20 df-3an 938 . . . . 5  |-  ( (
(/)  e.  dom  f  /\  Fun  G  /\  f  C_  G )  <->  ( ( (/) 
e.  dom  f  /\  Fun  G )  /\  f  C_  G ) )
21 3anrot 941 . . . . 5  |-  ( (
(/)  e.  dom  f  /\  Fun  G  /\  f  C_  G )  <->  ( Fun  G  /\  f  C_  G  /\  (/)  e.  dom  f
) )
22 ancom 438 . . . . 5  |-  ( ( ( (/)  e.  dom  f  /\  Fun  G )  /\  f  C_  G
)  <->  ( f  C_  G  /\  ( (/)  e.  dom  f  /\  Fun  G ) ) )
2320, 21, 223bitr3i 267 . . . 4  |-  ( ( Fun  G  /\  f  C_  G  /\  (/)  e.  dom  f )  <->  ( f  C_  G  /\  ( (/)  e.  dom  f  /\  Fun  G ) ) )
2419, 23sylibr 204 . . 3  |-  ( ( R  FrSe  A  /\  ta  /\  si )  -> 
( Fun  G  /\  f  C_  G  /\  (/)  e.  dom  f ) )
25 funssfv 5738 . . 3  |-  ( ( Fun  G  /\  f  C_  G  /\  (/)  e.  dom  f )  ->  ( G `  (/) )  =  ( f `  (/) ) )
2624, 25syl 16 . 2  |-  ( ( R  FrSe  A  /\  ta  /\  si )  -> 
( G `  (/) )  =  ( f `  (/) ) )
271simp2bi 973 . . 3  |-  ( ta 
->  ph' )
28273ad2ant2 979 . 2  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  ph' )
29 bnj545.1 . . . 4  |-  ( ph'  <->  (
f `  (/) )  = 
pred ( x ,  A ,  R ) )
30 eqtr 2452 . . . 4  |-  ( ( ( G `  (/) )  =  ( f `  (/) )  /\  ( f `  (/) )  = 
pred ( x ,  A ,  R ) )  ->  ( G `  (/) )  =  pred ( x ,  A ,  R ) )
3129, 30sylan2b 462 . . 3  |-  ( ( ( G `  (/) )  =  ( f `  (/) )  /\  ph' )  ->  ( G `  (/) )  =  pred ( x ,  A ,  R ) )
32 bnj545.7 . . 3  |-  ( ph"  <->  ( G `  (/) )  =  pred ( x ,  A ,  R ) )
3331, 32sylibr 204 . 2  |-  ( ( ( G `  (/) )  =  ( f `  (/) )  /\  ph' )  ->  ph" )
3426, 28, 33syl2anc 643 1  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  ph" )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    \ cdif 3309    u. cun 3310    C_ wss 3312   (/)c0 3620   {csn 3806   <.cop 3809   U_ciun 4085   suc csuc 4575   omcom 4837   dom cdm 4870   Fun wfun 5440    Fn wfn 5441   ` cfv 5446    predc-bnj14 28979    FrSe w-bnj15 28983
This theorem is referenced by:  bnj600  29217  bnj908  29229
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-res 4882  df-iota 5410  df-fun 5448  df-fn 5449  df-fv 5454
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