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Theorem bnj852 29269
Description: Technical lemma for bnj69 29356. 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 2811 . . . . . 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 29084 . . 3  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  E. x ( x  =  X  /\  ( R  FrSe  A  /\  X  e.  A ) ) )
5 eleq1 2356 . . . . . . 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 7360 . . . . . . . . . 10  |-  om  e.  _V
11 difexg 4178 . . . . . . . . . 10  |-  ( om  e.  _V  ->  ( om  \  { (/) } )  e.  _V )
1210, 11ax-mp 8 . . . . . . . . 9  |-  ( om 
\  { (/) } )  e.  _V
139, 12eqeltri 2366 . . . . . . . 8  |-  D  e. 
_V
14 zfregfr 7332 . . . . . . . 8  |-  _E  Fr  D
158, 13, 14bnj157 29207 . . . . . . 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 29228 . . . . . . . . 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 29268 . . . . . . . . 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 2535 . . . . . . . 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 2625 . . . . . 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 2643 . . . . . 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 29263 . . . . . . . . . 10  |-  ( x  =  X  ->  pred (
x ,  A ,  R )  =  pred ( X ,  A ,  R ) )
2827eqeq2d 2307 . . . . . . . . 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 2164 . . . . . 6  |-  ( x  =  X  ->  ( E! f ( f  Fn  n  /\  ( f `
 (/) )  =  pred ( x ,  A ,  R )  /\  ps ) 
<->  E! f ( f  Fn  n  /\  ph  /\ 
ps ) ) )
3332ralbidv 2576 . . . . 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 29090 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  E. x A. n  e.  D  E! f
( f  Fn  n  /\  ph  /\  ps )
)
3736bnj937 29119 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 1531    = wceq 1632    e. wcel 1696   E!weu 2156   A.wral 2556   _Vcvv 2801   [.wsbc 3004    \ cdif 3162   (/)c0 3468   {csn 3653   U_ciun 3921   class class class wbr 4039    _E cep 4319   suc csuc 4410   omcom 4672    Fn wfn 5266   ` cfv 5271   1oc1o 6488    predc-bnj14 29029    FrSe w-bnj15 29033
This theorem is referenced by:  bnj864  29270  bnj865  29271  bnj906  29278
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-reg 7322  ax-inf2 7358
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-1o 6495  df-bnj17 29028  df-bnj14 29030  df-bnj13 29032  df-bnj15 29034
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