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Theorem bnj849 29233
Description: Technical lemma for bnj69 29316. 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.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj849.1  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
bnj849.2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj849.3  |-  D  =  ( om  \  { (/)
} )
bnj849.4  |-  B  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
bnj849.5  |-  ( ch  <->  ( R  FrSe  A  /\  X  e.  A  /\  n  e.  D )
)
bnj849.6  |-  ( th  <->  ( f  Fn  n  /\  ph 
/\  ps ) )
bnj849.7  |-  ( ph'  <->  [. g  /  f ]. ph )
bnj849.8  |-  ( ps'  <->  [. g  /  f ]. ps )
bnj849.9  |-  ( th'  <->  [. g  / 
f ]. th )
bnj849.10  |-  ( ta  <->  ( R  FrSe  A  /\  X  e.  A )
)
Assertion
Ref Expression
bnj849  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  B  e.  _V )
Distinct variable groups:    A, f,
i, n, y    B, g    D, f, g, n    D, i    R, f, i, n, y    f, X, n    ch, f, g    ph, g    ps, g    ta, g, n    th, g
Allowed substitution hints:    ph( y, f, i, n)    ps( y,
f, i, n)    ch( y, i, n)    th( y,
f, i, n)    ta( y, f, i)    A( g)    B( y, f, i, n)    D( y)    R( g)    X( y, g, i)    ph'( y, f, g, i, n)    ps'( y, f, g, i, n)    th'( y, f, g, i, n)

Proof of Theorem bnj849
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 bnj849.10 . 2  |-  ( ta  <->  ( R  FrSe  A  /\  X  e.  A )
)
2 bnj849.1 . . . 4  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
3 bnj849.2 . . . 4  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
4 bnj849.3 . . . 4  |-  D  =  ( om  \  { (/)
} )
5 bnj849.5 . . . 4  |-  ( ch  <->  ( R  FrSe  A  /\  X  e.  A  /\  n  e.  D )
)
6 bnj849.6 . . . 4  |-  ( th  <->  ( f  Fn  n  /\  ph 
/\  ps ) )
72, 3, 4, 5, 6bnj865 29231 . . 3  |-  E. w A. n ( ch  ->  E. f  e.  w  th )
8 bnj849.4 . . . . . . . 8  |-  B  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
9 bnj849.7 . . . . . . . 8  |-  ( ph'  <->  [. g  /  f ]. ph )
10 bnj849.8 . . . . . . . 8  |-  ( ps'  <->  [. g  /  f ]. ps )
118, 9, 10bnj873 29232 . . . . . . 7  |-  B  =  { g  |  E. n  e.  D  (
g  Fn  n  /\  ph' 
/\  ps' ) }
12 df-rex 2703 . . . . . . . . 9  |-  ( E. n  e.  D  ( g  Fn  n  /\  ph' 
/\  ps' )  <->  E. n
( n  e.  D  /\  ( g  Fn  n  /\  ph'  /\  ps' ) ) )
13 19.29 1606 . . . . . . . . . . 11  |-  ( ( A. n ( ch 
->  E. f  e.  w  th )  /\  E. n
( n  e.  D  /\  ( g  Fn  n  /\  ph'  /\  ps' ) ) )  ->  E. n ( ( ch  ->  E. f  e.  w  th )  /\  ( n  e.  D  /\  ( g  Fn  n  /\  ph'  /\  ps' ) ) ) )
14 an12 773 . . . . . . . . . . . . 13  |-  ( ( ( ch  ->  E. f  e.  w  th )  /\  ( n  e.  D  /\  ( g  Fn  n  /\  ph'  /\  ps' ) ) )  <-> 
( n  e.  D  /\  ( ( ch  ->  E. f  e.  w  th )  /\  ( g  Fn  n  /\  ph'  /\  ps' ) ) ) )
15 df-3an 938 . . . . . . . . . . . . . . . 16  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  n  e.  D )  <->  ( ( R  FrSe  A  /\  X  e.  A
)  /\  n  e.  D ) )
161anbi1i 677 . . . . . . . . . . . . . . . 16  |-  ( ( ta  /\  n  e.  D )  <->  ( ( R  FrSe  A  /\  X  e.  A )  /\  n  e.  D ) )
1715, 5, 163bitr4i 269 . . . . . . . . . . . . . . 15  |-  ( ch  <->  ( ta  /\  n  e.  D ) )
18 id 20 . . . . . . . . . . . . . . . . 17  |-  ( ch 
->  ch )
19 bnj849.9 . . . . . . . . . . . . . . . . . . . 20  |-  ( th'  <->  [. g  / 
f ]. th )
206, 9, 10, 19bnj581 29216 . . . . . . . . . . . . . . . . . . . 20  |-  ( th'  <->  ( g  Fn  n  /\  ph'  /\  ps' ) )
2119, 20bitr3i 243 . . . . . . . . . . . . . . . . . . 19  |-  ( [. g  /  f ]. th  <->  ( g  Fn  n  /\  ph' 
/\  ps' ) )
222, 3, 4, 5, 6bnj864 29230 . . . . . . . . . . . . . . . . . . . 20  |-  ( ch 
->  E! f th )
23 df-rex 2703 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( E. f  e.  w  th  <->  E. f ( f  e.  w  /\  th )
)
24 exancom 1596 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( E. f ( f  e.  w  /\  th )  <->  E. f ( th  /\  f  e.  w )
)
2523, 24bitri 241 . . . . . . . . . . . . . . . . . . . . 21  |-  ( E. f  e.  w  th  <->  E. f ( th  /\  f  e.  w )
)
2625biimpi 187 . . . . . . . . . . . . . . . . . . . 20  |-  ( E. f  e.  w  th  ->  E. f ( th 
/\  f  e.  w
) )
27 nfeu1 2290 . . . . . . . . . . . . . . . . . . . . . . 23  |-  F/ f E! f th
28 nfe1 1747 . . . . . . . . . . . . . . . . . . . . . . 23  |-  F/ f E. f ( th 
/\  f  e.  w
)
2927, 28nfan 1846 . . . . . . . . . . . . . . . . . . . . . 22  |-  F/ f ( E! f th 
/\  E. f ( th 
/\  f  e.  w
) )
30 nfsbc1v 3172 . . . . . . . . . . . . . . . . . . . . . . 23  |-  F/ f
[. g  /  f ]. th
31 nfv 1629 . . . . . . . . . . . . . . . . . . . . . . 23  |-  F/ f  g  e.  w
3230, 31nfim 1832 . . . . . . . . . . . . . . . . . . . . . 22  |-  F/ f ( [. g  / 
f ]. th  ->  g  e.  w )
3329, 32nfim 1832 . . . . . . . . . . . . . . . . . . . . 21  |-  F/ f ( ( E! f th  /\  E. f
( th  /\  f  e.  w ) )  -> 
( [. g  /  f ]. th  ->  g  e.  w ) )
34 sbceq1a 3163 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( f  =  g  ->  ( th 
<-> 
[. g  /  f ]. th ) )
35 elequ1 1728 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( f  =  g  ->  (
f  e.  w  <->  g  e.  w ) )
3634, 35imbi12d 312 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( f  =  g  ->  (
( th  ->  f  e.  w )  <->  ( [. g  /  f ]. th  ->  g  e.  w ) ) )
3736imbi2d 308 . . . . . . . . . . . . . . . . . . . . 21  |-  ( f  =  g  ->  (
( ( E! f th  /\  E. f
( th  /\  f  e.  w ) )  -> 
( th  ->  f  e.  w ) )  <->  ( ( E! f th  /\  E. f ( th  /\  f  e.  w )
)  ->  ( [. g  /  f ]. th  ->  g  e.  w ) ) ) )
38 eupick 2343 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( E! f th  /\  E. f ( th  /\  f  e.  w )
)  ->  ( th  ->  f  e.  w ) )
3933, 37, 38chvar 1968 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( E! f th  /\  E. f ( th  /\  f  e.  w )
)  ->  ( [. g  /  f ]. th  ->  g  e.  w ) )
4022, 26, 39syl2an 464 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ch  /\  E. f  e.  w  th )  ->  ( [. g  / 
f ]. th  ->  g  e.  w ) )
4121, 40syl5bir 210 . . . . . . . . . . . . . . . . . 18  |-  ( ( ch  /\  E. f  e.  w  th )  ->  ( ( g  Fn  n  /\  ph'  /\  ps' )  -> 
g  e.  w ) )
4241ex 424 . . . . . . . . . . . . . . . . 17  |-  ( ch 
->  ( E. f  e.  w  th  ->  (
( g  Fn  n  /\  ph'  /\  ps' )  ->  g  e.  w ) ) )
4318, 42embantd 52 . . . . . . . . . . . . . . . 16  |-  ( ch 
->  ( ( ch  ->  E. f  e.  w  th )  ->  ( ( g  Fn  n  /\  ph'  /\  ps' )  -> 
g  e.  w ) ) )
4443imp3a 421 . . . . . . . . . . . . . . 15  |-  ( ch 
->  ( ( ( ch 
->  E. f  e.  w  th )  /\  (
g  Fn  n  /\  ph' 
/\  ps' ) )  -> 
g  e.  w ) )
4517, 44sylbir 205 . . . . . . . . . . . . . 14  |-  ( ( ta  /\  n  e.  D )  ->  (
( ( ch  ->  E. f  e.  w  th )  /\  ( g  Fn  n  /\  ph'  /\  ps' ) )  ->  g  e.  w
) )
4645expimpd 587 . . . . . . . . . . . . 13  |-  ( ta 
->  ( ( n  e.  D  /\  ( ( ch  ->  E. f  e.  w  th )  /\  ( g  Fn  n  /\  ph'  /\  ps' ) ) )  ->  g  e.  w
) )
4714, 46syl5bi 209 . . . . . . . . . . . 12  |-  ( ta 
->  ( ( ( ch 
->  E. f  e.  w  th )  /\  (
n  e.  D  /\  ( g  Fn  n  /\  ph'  /\  ps' ) ) )  ->  g  e.  w
) )
4847exlimdv 1646 . . . . . . . . . . 11  |-  ( ta 
->  ( E. n ( ( ch  ->  E. f  e.  w  th )  /\  ( n  e.  D  /\  ( g  Fn  n  /\  ph'  /\  ps' ) ) )  ->  g  e.  w
) )
4913, 48syl5 30 . . . . . . . . . 10  |-  ( ta 
->  ( ( A. n
( ch  ->  E. f  e.  w  th )  /\  E. n ( n  e.  D  /\  (
g  Fn  n  /\  ph' 
/\  ps' ) ) )  ->  g  e.  w
) )
5049expdimp 427 . . . . . . . . 9  |-  ( ( ta  /\  A. n
( ch  ->  E. f  e.  w  th )
)  ->  ( E. n ( n  e.  D  /\  ( g  Fn  n  /\  ph'  /\  ps' ) )  ->  g  e.  w
) )
5112, 50syl5bi 209 . . . . . . . 8  |-  ( ( ta  /\  A. n
( ch  ->  E. f  e.  w  th )
)  ->  ( E. n  e.  D  (
g  Fn  n  /\  ph' 
/\  ps' )  ->  g  e.  w ) )
5251abssdv 3409 . . . . . . 7  |-  ( ( ta  /\  A. n
( ch  ->  E. f  e.  w  th )
)  ->  { g  |  E. n  e.  D  ( g  Fn  n  /\  ph'  /\  ps' ) }  C_  w )
5311, 52syl5eqss 3384 . . . . . 6  |-  ( ( ta  /\  A. n
( ch  ->  E. f  e.  w  th )
)  ->  B  C_  w
)
54 vex 2951 . . . . . . 7  |-  w  e. 
_V
5554ssex 4339 . . . . . 6  |-  ( B 
C_  w  ->  B  e.  _V )
5653, 55syl 16 . . . . 5  |-  ( ( ta  /\  A. n
( ch  ->  E. f  e.  w  th )
)  ->  B  e.  _V )
5756ex 424 . . . 4  |-  ( ta 
->  ( A. n ( ch  ->  E. f  e.  w  th )  ->  B  e.  _V )
)
5857exlimdv 1646 . . 3  |-  ( ta 
->  ( E. w A. n ( ch  ->  E. f  e.  w  th )  ->  B  e.  _V ) )
597, 58mpi 17 . 2  |-  ( ta 
->  B  e.  _V )
601, 59sylbir 205 1  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  B  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   A.wal 1549   E.wex 1550    = wceq 1652    e. wcel 1725   E!weu 2280   {cab 2421   A.wral 2697   E.wrex 2698   _Vcvv 2948   [.wsbc 3153    \ cdif 3309    C_ wss 3312   (/)c0 3620   {csn 3806   U_ciun 4085   suc csuc 4575   omcom 4837    Fn wfn 5441   ` cfv 5446    predc-bnj14 28989    FrSe w-bnj15 28993
This theorem is referenced by:  bnj893  29236
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-reg 7552  ax-inf2 7588
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-1o 6716  df-bnj17 28988  df-bnj14 28990  df-bnj13 28992  df-bnj15 28994
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