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Theorem bnj545 29699
Description: Technical lemma for bnj852 29725. 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 1022 . . . . . . . . 9  |-  ( ta 
->  f  Fn  m
)
3 bnj545.5 . . . . . . . . . 10  |-  ( si  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  m )
)
43simp1bi 1022 . . . . . . . . 9  |-  ( si  ->  m  e.  D )
52, 4anim12i 569 . . . . . . . 8  |-  ( ( ta  /\  si )  ->  ( f  Fn  m  /\  m  e.  D
) )
653adant1 1025 . . . . . . 7  |-  ( ( R  FrSe  A  /\  ta  /\  si )  -> 
( f  Fn  m  /\  m  e.  D
) )
7 bnj545.2 . . . . . . . . 9  |-  D  =  ( om  \  { (/)
} )
87bnj529 29544 . . . . . . . 8  |-  ( m  e.  D  ->  (/)  e.  m
)
9 fndm 5673 . . . . . . . 8  |-  ( f  Fn  m  ->  dom  f  =  m )
10 eleq2 2517 . . . . . . . . 9  |-  ( dom  f  =  m  -> 
( (/)  e.  dom  f  <->  (/)  e.  m ) )
1110biimparc 490 . . . . . . . 8  |-  ( (
(/)  e.  m  /\  dom  f  =  m
)  ->  (/)  e.  dom  f )
128, 9, 11syl2anr 481 . . . . . . 7  |-  ( ( f  Fn  m  /\  m  e.  D )  -> 
(/)  e.  dom  f )
136, 12syl 17 . . . . . 6  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  (/) 
e.  dom  f )
14 bnj545.6 . . . . . . 7  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  G  Fn  n )
1514bnj930 29574 . . . . . 6  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  Fun  G )
1613, 15jca 535 . . . . 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 29575 . . . . 5  |-  f  C_  G
1916, 18jctil 540 . . . 4  |-  ( ( R  FrSe  A  /\  ta  /\  si )  -> 
( f  C_  G  /\  ( (/)  e.  dom  f  /\  Fun  G ) ) )
20 df-3an 986 . . . . 5  |-  ( (
(/)  e.  dom  f  /\  Fun  G  /\  f  C_  G )  <->  ( ( (/) 
e.  dom  f  /\  Fun  G )  /\  f  C_  G ) )
21 3anrot 989 . . . . 5  |-  ( (
(/)  e.  dom  f  /\  Fun  G  /\  f  C_  G )  <->  ( Fun  G  /\  f  C_  G  /\  (/)  e.  dom  f
) )
22 ancom 452 . . . . 5  |-  ( ( ( (/)  e.  dom  f  /\  Fun  G )  /\  f  C_  G
)  <->  ( f  C_  G  /\  ( (/)  e.  dom  f  /\  Fun  G ) ) )
2320, 21, 223bitr3i 279 . . . 4  |-  ( ( Fun  G  /\  f  C_  G  /\  (/)  e.  dom  f )  <->  ( f  C_  G  /\  ( (/)  e.  dom  f  /\  Fun  G ) ) )
2419, 23sylibr 216 . . 3  |-  ( ( R  FrSe  A  /\  ta  /\  si )  -> 
( Fun  G  /\  f  C_  G  /\  (/)  e.  dom  f ) )
25 funssfv 5878 . . 3  |-  ( ( Fun  G  /\  f  C_  G  /\  (/)  e.  dom  f )  ->  ( G `  (/) )  =  ( f `  (/) ) )
2624, 25syl 17 . 2  |-  ( ( R  FrSe  A  /\  ta  /\  si )  -> 
( G `  (/) )  =  ( f `  (/) ) )
271simp2bi 1023 . . 3  |-  ( ta 
->  ph' )
28273ad2ant2 1029 . 2  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  ph' )
29 bnj545.1 . . . 4  |-  ( ph'  <->  (
f `  (/) )  = 
pred ( x ,  A ,  R ) )
30 eqtr 2469 . . . 4  |-  ( ( ( G `  (/) )  =  ( f `  (/) )  /\  ( f `  (/) )  = 
pred ( x ,  A ,  R ) )  ->  ( G `  (/) )  =  pred ( x ,  A ,  R ) )
3129, 30sylan2b 478 . . 3  |-  ( ( ( G `  (/) )  =  ( f `  (/) )  /\  ph' )  ->  ( G `  (/) )  =  pred ( x ,  A ,  R ) )
32 bnj545.7 . . 3  |-  ( ph"  <->  ( G `  (/) )  =  pred ( x ,  A ,  R ) )
3331, 32sylibr 216 . 2  |-  ( ( ( G `  (/) )  =  ( f `  (/) )  /\  ph' )  ->  ph" )
3426, 28, 33syl2anc 666 1  |-  ( ( R  FrSe  A  /\  ta  /\  si )  ->  ph" )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 984    = wceq 1443    e. wcel 1886    \ cdif 3400    u. cun 3401    C_ wss 3403   (/)c0 3730   {csn 3967   <.cop 3973   U_ciun 4277   dom cdm 4833   suc csuc 5424   Fun wfun 5575    Fn wfn 5576   ` cfv 5581   omcom 6689    predc-bnj14 29486    FrSe w-bnj15 29490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pr 4638  ax-un 6580
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-sbc 3267  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-br 4402  df-opab 4461  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-res 4845  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-fv 5589  df-om 6690
This theorem is referenced by:  bnj600  29723  bnj908  29735
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