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Theorem bnj852 32010
Description: Technical lemma for bnj69 32097. 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 3004 . . . . . 6  |-  ( X  e.  A  ->  E. x  x  =  X )
21adantl 466 . . . . 5  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  E. x  x  =  X )
32ancri 552 . . . 4  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  ( E. x  x  =  X  /\  ( R  FrSe  A  /\  X  e.  A ) ) )
43bnj534 31827 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  E. x ( x  =  X  /\  ( R  FrSe  A  /\  X  e.  A ) ) )
5 eleq1 2503 . . . . . . 7  |-  ( x  =  X  ->  (
x  e.  A  <->  X  e.  A ) )
65anbi2d 703 . . . . . 6  |-  ( x  =  X  ->  (
( R  FrSe  A  /\  x  e.  A
)  <->  ( R  FrSe  A  /\  X  e.  A
) ) )
76biimpar 485 . . . . 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 7870 . . . . . . . . . 10  |-  om  e.  _V
11 difexg 4461 . . . . . . . . . 10  |-  ( om  e.  _V  ->  ( om  \  { (/) } )  e.  _V )
1210, 11ax-mp 5 . . . . . . . . 9  |-  ( om 
\  { (/) } )  e.  _V
139, 12eqeltri 2513 . . . . . . . 8  |-  D  e. 
_V
14 zfregfr 7839 . . . . . . . 8  |-  _E  Fr  D
158, 13, 14bnj157 31948 . . . . . . 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 31969 . . . . . . . . 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 32009 . . . . . . . . 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 2711 . . . . . . . 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 434 . . . . . . 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 2806 . . . . . 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 2824 . . . . . 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 32004 . . . . . . . . . 10  |-  ( x  =  X  ->  pred (
x ,  A ,  R )  =  pred ( X ,  A ,  R ) )
2827eqeq2d 2454 . . . . . . . . 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 1294 . . . . . . 7  |-  ( x  =  X  ->  (
( f  Fn  n  /\  ( f `  (/) )  = 
pred ( x ,  A ,  R )  /\  ps )  <->  ( f  Fn  n  /\  ph  /\  ps ) ) )
3231eubidv 2276 . . . . . 6  |-  ( x  =  X  ->  ( E! f ( f  Fn  n  /\  ( f `
 (/) )  =  pred ( x ,  A ,  R )  /\  ps ) 
<->  E! f ( f  Fn  n  /\  ph  /\ 
ps ) ) )
3332ralbidv 2756 . . . . 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 465 . . . 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 31833 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  E. x A. n  e.  D  E! f
( f  Fn  n  /\  ph  /\  ps )
)
3736bnj937 31861 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 369    /\ w3a 965    = wceq 1369   E.wex 1586    e. wcel 1756   E!weu 2253   A.wral 2736   _Vcvv 2993   [.wsbc 3207    \ cdif 3346   (/)c0 3658   {csn 3898   U_ciun 4192   class class class wbr 4313    _E cep 4651   suc csuc 4742    Fn wfn 5434   ` cfv 5439   omcom 6497   1oc1o 6934    predc-bnj14 31772    FrSe w-bnj15 31776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-reg 7828  ax-inf2 7868
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-om 6498  df-1o 6941  df-bnj17 31771  df-bnj14 31773  df-bnj13 31775  df-bnj15 31777
This theorem is referenced by:  bnj864  32011  bnj865  32012  bnj906  32019
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