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Theorem bnj852 34380
Description: Technical lemma for bnj69 34467. 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 3117 . . . . . 6  |-  ( X  e.  A  ->  E. x  x  =  X )
21adantl 464 . . . . 5  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  E. x  x  =  X )
32ancri 550 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  ( E. x  x  =  X  /\  ( R  FrSe  A  /\  X  e.  A ) ) )
43bnj534 34196 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  E. x ( x  =  X  /\  ( R  FrSe  A  /\  X  e.  A ) ) )
5 eleq1 2526 . . . . . . 7  |-  ( x  =  X  ->  (
x  e.  A  <->  X  e.  A ) )
65anbi2d 701 . . . . . 6  |-  ( x  =  X  ->  (
( R  FrSe  A  /\  x  e.  A
)  <->  ( R  FrSe  A  /\  X  e.  A
) ) )
76biimpar 483 . . . . 5  |-  ( ( x  =  X  /\  ( R  FrSe  A  /\  X  e.  A )
)  ->  ( R  FrSe  A  /\  x  e.  A ) )
8 biid 236 . . . . . . . 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 8051 . . . . . . . . . 10  |-  om  e.  _V
11 difexg 4585 . . . . . . . . . 10  |-  ( om  e.  _V  ->  ( om  \  { (/) } )  e.  _V )
1210, 11ax-mp 5 . . . . . . . . 9  |-  ( om 
\  { (/) } )  e.  _V
139, 12eqeltri 2538 . . . . . . . 8  |-  D  e. 
_V
14 zfregfr 8020 . . . . . . . 8  |-  _E  Fr  D
158, 13, 14bnj157 34318 . . . . . . 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 236 . . . . . . . . . 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 236 . . . . . . . . . 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 34339 . . . . . . . . 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 34379 . . . . . . . . 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 2767 . . . . . . . 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 432 . . . . . . 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 2817 . . . . . 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 2859 . . . . . 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 208 . . . . 5  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  A. n  e.  D  E! f ( f  Fn  n  /\  ( f `
 (/) )  =  pred ( x ,  A ,  R )  /\  ps ) )
267, 25syl 16 . . . 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 34374 . . . . . . . . . 10  |-  ( x  =  X  ->  pred (
x ,  A ,  R )  =  pred ( X ,  A ,  R ) )
2827eqeq2d 2468 . . . . . . . . 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 263 . . . . . . . 8  |-  ( x  =  X  ->  (
( f `  (/) )  = 
pred ( x ,  A ,  R )  <->  ph ) )
31303anbi2d 1302 . . . . . . 7  |-  ( x  =  X  ->  (
( f  Fn  n  /\  ( f `  (/) )  = 
pred ( x ,  A ,  R )  /\  ps )  <->  ( f  Fn  n  /\  ph  /\  ps ) ) )
3231eubidv 2306 . . . . . 6  |-  ( x  =  X  ->  ( E! f ( f  Fn  n  /\  ( f `
 (/) )  =  pred ( x ,  A ,  R )  /\  ps ) 
<->  E! f ( f  Fn  n  /\  ph  /\ 
ps ) ) )
3332ralbidv 2893 . . . . 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 463 . . . 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 210 . . 3  |-  ( ( x  =  X  /\  ( R  FrSe  A  /\  X  e.  A )
)  ->  A. n  e.  D  E! f
( f  Fn  n  /\  ph  /\  ps )
)
364, 35bnj593 34203 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  E. x A. n  e.  D  E! f
( f  Fn  n  /\  ph  /\  ps )
)
3736bnj937 34231 1  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  A. n  e.  D  E! f ( f  Fn  n  /\  ph  /\  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398   E.wex 1617    e. wcel 1823   E!weu 2284   A.wral 2804   _Vcvv 3106   [.wsbc 3324    \ cdif 3458   (/)c0 3783   {csn 4016   U_ciun 4315   class class class wbr 4439    _E cep 4778   suc csuc 4869    Fn wfn 5565   ` cfv 5570   omcom 6673   1oc1o 7115    predc-bnj14 34141    FrSe w-bnj15 34145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-reg 8010  ax-inf2 8049
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-om 6674  df-1o 7122  df-bnj17 34140  df-bnj14 34142  df-bnj13 34144  df-bnj15 34146
This theorem is referenced by:  bnj864  34381  bnj865  34382  bnj906  34389
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