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Theorem bnj852 28953
Description: Technical lemma for bnj69 29040. 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
bnj852.1  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
bnj852.2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj852.3  |-  D  =  ( om  \  { (/)
} )
Assertion
Ref Expression
bnj852  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  A. n  e.  D  E! f ( f  Fn  n  /\  ph  /\  ps ) )
Distinct variable groups:    A, f,
i, n, y    D, f, i, n    R, f, i, n, y    f, X, n
Allowed substitution hints:    ph( y, f, i, n)    ps( y,
f, i, n)    D( y)    X( y, i)

Proof of Theorem bnj852
Dummy variables  x  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elisset 2798 . . . . . 6  |-  ( X  e.  A  ->  E. x  x  =  X )
21adantl 452 . . . . 5  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  E. x  x  =  X )
32ancri 535 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  ( E. x  x  =  X  /\  ( R  FrSe  A  /\  X  e.  A ) ) )
43bnj534 28768 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  E. x ( x  =  X  /\  ( R  FrSe  A  /\  X  e.  A ) ) )
5 eleq1 2343 . . . . . . 7  |-  ( x  =  X  ->  (
x  e.  A  <->  X  e.  A ) )
65anbi2d 684 . . . . . 6  |-  ( x  =  X  ->  (
( R  FrSe  A  /\  x  e.  A
)  <->  ( R  FrSe  A  /\  X  e.  A
) ) )
76biimpar 471 . . . . 5  |-  ( ( x  =  X  /\  ( R  FrSe  A  /\  X  e.  A )
)  ->  ( R  FrSe  A  /\  x  e.  A ) )
8 biid 227 . . . . . . . 8  |-  ( A. z  e.  D  (
z  _E  n  ->  [. z  /  n ]. ( ( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  n  /\  ( f `  (/) )  = 
pred ( x ,  A ,  R )  /\  ps ) ) )  <->  A. z  e.  D  ( z  _E  n  ->  [. z  /  n ]. ( ( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  n  /\  ( f `  (/) )  = 
pred ( x ,  A ,  R )  /\  ps ) ) ) )
9 bnj852.3 . . . . . . . . 9  |-  D  =  ( om  \  { (/)
} )
10 omex 7344 . . . . . . . . . 10  |-  om  e.  _V
11 difexg 4162 . . . . . . . . . 10  |-  ( om  e.  _V  ->  ( om  \  { (/) } )  e.  _V )
1210, 11ax-mp 8 . . . . . . . . 9  |-  ( om 
\  { (/) } )  e.  _V
139, 12eqeltri 2353 . . . . . . . 8  |-  D  e. 
_V
14 zfregfr 7316 . . . . . . . 8  |-  _E  Fr  D
158, 13, 14bnj157 28891 . . . . . . 7  |-  ( A. n  e.  D  ( A. z  e.  D  ( z  _E  n  ->  [. z  /  n ]. ( ( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  n  /\  ( f `  (/) )  = 
pred ( x ,  A ,  R )  /\  ps ) ) )  ->  ( ( R  FrSe  A  /\  x  e.  A )  ->  E! f ( f  Fn  n  /\  ( f `
 (/) )  =  pred ( x ,  A ,  R )  /\  ps ) ) )  ->  A. n  e.  D  ( ( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  n  /\  ( f `  (/) )  = 
pred ( x ,  A ,  R )  /\  ps ) ) )
16 biid 227 . . . . . . . . . 10  |-  ( ( f `  (/) )  = 
pred ( x ,  A ,  R )  <-> 
( f `  (/) )  = 
pred ( x ,  A ,  R ) )
17 bnj852.2 . . . . . . . . . 10  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
18 biid 227 . . . . . . . . . 10  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  n  /\  ( f `  (/) )  = 
pred ( x ,  A ,  R )  /\  ps ) )  <-> 
( ( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  n  /\  ( f `  (/) )  = 
pred ( x ,  A ,  R )  /\  ps ) ) )
1916, 17, 9, 18, 8bnj153 28912 . . . . . . . . 9  |-  ( n  =  1o  ->  (
( n  e.  D  /\  A. z  e.  D  ( z  _E  n  ->  [. z  /  n ]. ( ( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  n  /\  ( f `  (/) )  = 
pred ( x ,  A ,  R )  /\  ps ) ) ) )  ->  (
( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  n  /\  ( f `  (/) )  = 
pred ( x ,  A ,  R )  /\  ps ) ) ) )
2016, 17, 9, 18, 8bnj601 28952 . . . . . . . . 9  |-  ( n  =/=  1o  ->  (
( n  e.  D  /\  A. z  e.  D  ( z  _E  n  ->  [. z  /  n ]. ( ( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  n  /\  ( f `  (/) )  = 
pred ( x ,  A ,  R )  /\  ps ) ) ) )  ->  (
( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  n  /\  ( f `  (/) )  = 
pred ( x ,  A ,  R )  /\  ps ) ) ) )
2119, 20pm2.61ine 2522 . . . . . . . 8  |-  ( ( n  e.  D  /\  A. z  e.  D  ( z  _E  n  ->  [. z  /  n ]. ( ( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  n  /\  ( f `  (/) )  = 
pred ( x ,  A ,  R )  /\  ps ) ) ) )  ->  (
( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  n  /\  ( f `  (/) )  = 
pred ( x ,  A ,  R )  /\  ps ) ) )
2221ex 423 . . . . . . 7  |-  ( n  e.  D  ->  ( A. z  e.  D  ( z  _E  n  ->  [. z  /  n ]. ( ( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  n  /\  ( f `  (/) )  = 
pred ( x ,  A ,  R )  /\  ps ) ) )  ->  ( ( R  FrSe  A  /\  x  e.  A )  ->  E! f ( f  Fn  n  /\  ( f `
 (/) )  =  pred ( x ,  A ,  R )  /\  ps ) ) ) )
2315, 22mprg 2612 . . . . . 6  |-  A. n  e.  D  ( ( R  FrSe  A  /\  x  e.  A )  ->  E! f ( f  Fn  n  /\  ( f `
 (/) )  =  pred ( x ,  A ,  R )  /\  ps ) )
24 r19.21v 2630 . . . . . 6  |-  ( A. n  e.  D  (
( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  n  /\  ( f `  (/) )  = 
pred ( x ,  A ,  R )  /\  ps ) )  <-> 
( ( R  FrSe  A  /\  x  e.  A
)  ->  A. n  e.  D  E! f
( f  Fn  n  /\  ( f `  (/) )  = 
pred ( x ,  A ,  R )  /\  ps ) ) )
2523, 24mpbi 199 . . . . 5  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  A. n  e.  D  E! f ( f  Fn  n  /\  ( f `
 (/) )  =  pred ( x ,  A ,  R )  /\  ps ) )
267, 25syl 15 . . . 4  |-  ( ( x  =  X  /\  ( R  FrSe  A  /\  X  e.  A )
)  ->  A. n  e.  D  E! f
( f  Fn  n  /\  ( f `  (/) )  = 
pred ( x ,  A ,  R )  /\  ps ) )
27 bnj602 28947 . . . . . . . . . 10  |-  ( x  =  X  ->  pred (
x ,  A ,  R )  =  pred ( X ,  A ,  R ) )
2827eqeq2d 2294 . . . . . . . . 9  |-  ( x  =  X  ->  (
( f `  (/) )  = 
pred ( x ,  A ,  R )  <-> 
( f `  (/) )  = 
pred ( X ,  A ,  R )
) )
29 bnj852.1 . . . . . . . . 9  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
3028, 29syl6bbr 254 . . . . . . . 8  |-  ( x  =  X  ->  (
( f `  (/) )  = 
pred ( x ,  A ,  R )  <->  ph ) )
31303anbi2d 1257 . . . . . . 7  |-  ( x  =  X  ->  (
( f  Fn  n  /\  ( f `  (/) )  = 
pred ( x ,  A ,  R )  /\  ps )  <->  ( f  Fn  n  /\  ph  /\  ps ) ) )
3231eubidv 2151 . . . . . 6  |-  ( x  =  X  ->  ( E! f ( f  Fn  n  /\  ( f `
 (/) )  =  pred ( x ,  A ,  R )  /\  ps ) 
<->  E! f ( f  Fn  n  /\  ph  /\ 
ps ) ) )
3332ralbidv 2563 . . . . 5  |-  ( x  =  X  ->  ( A. n  e.  D  E! f ( f  Fn  n  /\  ( f `
 (/) )  =  pred ( x ,  A ,  R )  /\  ps ) 
<-> 
A. n  e.  D  E! f ( f  Fn  n  /\  ph  /\  ps ) ) )
3433adantr 451 . . . 4  |-  ( ( x  =  X  /\  ( R  FrSe  A  /\  X  e.  A )
)  ->  ( A. n  e.  D  E! f ( f  Fn  n  /\  ( f `
 (/) )  =  pred ( x ,  A ,  R )  /\  ps ) 
<-> 
A. n  e.  D  E! f ( f  Fn  n  /\  ph  /\  ps ) ) )
3526, 34mpbid 201 . . 3  |-  ( ( x  =  X  /\  ( R  FrSe  A  /\  X  e.  A )
)  ->  A. n  e.  D  E! f
( f  Fn  n  /\  ph  /\  ps )
)
364, 35bnj593 28774 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  E. x A. n  e.  D  E! f
( f  Fn  n  /\  ph  /\  ps )
)
3736bnj937 28803 1  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  A. n  e.  D  E! f ( f  Fn  n  /\  ph  /\  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684   E!weu 2143   A.wral 2543   _Vcvv 2788   [.wsbc 2991    \ cdif 3149   (/)c0 3455   {csn 3640   U_ciun 3905   class class class wbr 4023    _E cep 4303   suc csuc 4394   omcom 4656    Fn wfn 5250   ` cfv 5255   1oc1o 6472    predc-bnj14 28713    FrSe w-bnj15 28717
This theorem is referenced by:  bnj864  28954  bnj865  28955  bnj906  28962
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-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-reg 7306  ax-inf2 7342
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-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  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-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  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-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1o 6479  df-bnj17 28712  df-bnj14 28714  df-bnj13 28716  df-bnj15 28718
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