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Theorem bnj852 29732
Description: Technical lemma for bnj69 29819. 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 3057 . . . . . 6  |-  ( X  e.  A  ->  E. x  x  =  X )
21adantl 468 . . . . 5  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  E. x  x  =  X )
32ancri 555 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  ( E. x  x  =  X  /\  ( R  FrSe  A  /\  X  e.  A ) ) )
43bnj534 29548 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  E. x ( x  =  X  /\  ( R  FrSe  A  /\  X  e.  A ) ) )
5 eleq1 2517 . . . . . . 7  |-  ( x  =  X  ->  (
x  e.  A  <->  X  e.  A ) )
65anbi2d 710 . . . . . 6  |-  ( x  =  X  ->  (
( R  FrSe  A  /\  x  e.  A
)  <->  ( R  FrSe  A  /\  X  e.  A
) ) )
76biimpar 488 . . . . 5  |-  ( ( x  =  X  /\  ( R  FrSe  A  /\  X  e.  A )
)  ->  ( R  FrSe  A  /\  x  e.  A ) )
8 biid 240 . . . . . . . 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 8148 . . . . . . . . . 10  |-  om  e.  _V
11 difexg 4551 . . . . . . . . . 10  |-  ( om  e.  _V  ->  ( om  \  { (/) } )  e.  _V )
1210, 11ax-mp 5 . . . . . . . . 9  |-  ( om 
\  { (/) } )  e.  _V
139, 12eqeltri 2525 . . . . . . . 8  |-  D  e. 
_V
14 zfregfr 8117 . . . . . . . 8  |-  _E  Fr  D
158, 13, 14bnj157 29670 . . . . . . 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 240 . . . . . . . . . 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 240 . . . . . . . . . 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 29691 . . . . . . . . 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 29731 . . . . . . . . 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 2707 . . . . . . . 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 436 . . . . . . 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 2751 . . . . . 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 2793 . . . . . 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 212 . . . . 5  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  A. n  e.  D  E! f ( f  Fn  n  /\  ( f `
 (/) )  =  pred ( x ,  A ,  R )  /\  ps ) )
267, 25syl 17 . . . 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 29726 . . . . . . . . . 10  |-  ( x  =  X  ->  pred (
x ,  A ,  R )  =  pred ( X ,  A ,  R ) )
2827eqeq2d 2461 . . . . . . . . 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 267 . . . . . . . 8  |-  ( x  =  X  ->  (
( f `  (/) )  = 
pred ( x ,  A ,  R )  <->  ph ) )
31303anbi2d 1344 . . . . . . 7  |-  ( x  =  X  ->  (
( f  Fn  n  /\  ( f `  (/) )  = 
pred ( x ,  A ,  R )  /\  ps )  <->  ( f  Fn  n  /\  ph  /\  ps ) ) )
3231eubidv 2319 . . . . . 6  |-  ( x  =  X  ->  ( E! f ( f  Fn  n  /\  ( f `
 (/) )  =  pred ( x ,  A ,  R )  /\  ps ) 
<->  E! f ( f  Fn  n  /\  ph  /\ 
ps ) ) )
3332ralbidv 2827 . . . . 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 467 . . . 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 214 . . 3  |-  ( ( x  =  X  /\  ( R  FrSe  A  /\  X  e.  A )
)  ->  A. n  e.  D  E! f
( f  Fn  n  /\  ph  /\  ps )
)
364, 35bnj593 29555 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  E. x A. n  e.  D  E! f
( f  Fn  n  /\  ph  /\  ps )
)
3736bnj937 29583 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 188    /\ wa 371    /\ w3a 985    = wceq 1444   E.wex 1663    e. wcel 1887   E!weu 2299   A.wral 2737   _Vcvv 3045   [.wsbc 3267    \ cdif 3401   (/)c0 3731   {csn 3968   U_ciun 4278   class class class wbr 4402    _E cep 4743   suc csuc 5425    Fn wfn 5577   ` cfv 5582   omcom 6692   1oc1o 7175    predc-bnj14 29493    FrSe w-bnj15 29497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-reg 8107  ax-inf2 8146
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-om 6693  df-1o 7182  df-bnj17 29492  df-bnj14 29494  df-bnj13 29496  df-bnj15 29498
This theorem is referenced by:  bnj864  29733  bnj865  29734  bnj906  29741
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