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Theorem bnj849 33080
Description: Technical lemma for bnj69 33163. 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 33078 . . 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 33079 . . . . . . 7  |-  B  =  { g  |  E. n  e.  D  (
g  Fn  n  /\  ph' 
/\  ps' ) }
12 df-rex 2820 . . . . . . . . 9  |-  ( E. n  e.  D  ( g  Fn  n  /\  ph' 
/\  ps' )  <->  E. n
( n  e.  D  /\  ( g  Fn  n  /\  ph'  /\  ps' ) ) )
13 19.29 1660 . . . . . . . . . . 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 795 . . . . . . . . . . . . 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 975 . . . . . . . . . . . . . . . 16  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  n  e.  D )  <->  ( ( R  FrSe  A  /\  X  e.  A
)  /\  n  e.  D ) )
161anbi1i 695 . . . . . . . . . . . . . . . 16  |-  ( ( ta  /\  n  e.  D )  <->  ( ( R  FrSe  A  /\  X  e.  A )  /\  n  e.  D ) )
1715, 5, 163bitr4i 277 . . . . . . . . . . . . . . 15  |-  ( ch  <->  ( ta  /\  n  e.  D ) )
18 id 22 . . . . . . . . . . . . . . . . 17  |-  ( ch 
->  ch )
19 bnj849.9 . . . . . . . . . . . . . . . . . . . 20  |-  ( th'  <->  [. g  / 
f ]. th )
206, 9, 10, 19bnj581 33063 . . . . . . . . . . . . . . . . . . . 20  |-  ( th'  <->  ( g  Fn  n  /\  ph'  /\  ps' ) )
2119, 20bitr3i 251 . . . . . . . . . . . . . . . . . . 19  |-  ( [. g  /  f ]. th  <->  ( g  Fn  n  /\  ph' 
/\  ps' ) )
222, 3, 4, 5, 6bnj864 33077 . . . . . . . . . . . . . . . . . . . 20  |-  ( ch 
->  E! f th )
23 df-rex 2820 . . . . . . . . . . . . . . . . . . . . 21  |-  ( E. f  e.  w  th  <->  E. f ( f  e.  w  /\  th )
)
24 exancom 1648 . . . . . . . . . . . . . . . . . . . . 21  |-  ( E. f ( f  e.  w  /\  th )  <->  E. f ( th  /\  f  e.  w )
)
2523, 24sylbb 197 . . . . . . . . . . . . . . . . . . . 20  |-  ( E. f  e.  w  th  ->  E. f ( th 
/\  f  e.  w
) )
26 nfeu1 2288 . . . . . . . . . . . . . . . . . . . . . . 23  |-  F/ f E! f th
27 nfe1 1789 . . . . . . . . . . . . . . . . . . . . . . 23  |-  F/ f E. f ( th 
/\  f  e.  w
)
2826, 27nfan 1875 . . . . . . . . . . . . . . . . . . . . . 22  |-  F/ f ( E! f th 
/\  E. f ( th 
/\  f  e.  w
) )
29 nfsbc1v 3351 . . . . . . . . . . . . . . . . . . . . . . 23  |-  F/ f
[. g  /  f ]. th
30 nfv 1683 . . . . . . . . . . . . . . . . . . . . . . 23  |-  F/ f  g  e.  w
3129, 30nfim 1867 . . . . . . . . . . . . . . . . . . . . . 22  |-  F/ f ( [. g  / 
f ]. th  ->  g  e.  w )
3228, 31nfim 1867 . . . . . . . . . . . . . . . . . . . . 21  |-  F/ f ( ( E! f th  /\  E. f
( th  /\  f  e.  w ) )  -> 
( [. g  /  f ]. th  ->  g  e.  w ) )
33 sbceq1a 3342 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( f  =  g  ->  ( th 
<-> 
[. g  /  f ]. th ) )
34 elequ1 1770 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( f  =  g  ->  (
f  e.  w  <->  g  e.  w ) )
3533, 34imbi12d 320 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( f  =  g  ->  (
( th  ->  f  e.  w )  <->  ( [. g  /  f ]. th  ->  g  e.  w ) ) )
3635imbi2d 316 . . . . . . . . . . . . . . . . . . . . 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 ) ) ) )
37 eupick 2364 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( E! f th  /\  E. f ( th  /\  f  e.  w )
)  ->  ( th  ->  f  e.  w ) )
3832, 36, 37chvar 1982 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( E! f th  /\  E. f ( th  /\  f  e.  w )
)  ->  ( [. g  /  f ]. th  ->  g  e.  w ) )
3922, 25, 38syl2an 477 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ch  /\  E. f  e.  w  th )  ->  ( [. g  / 
f ]. th  ->  g  e.  w ) )
4021, 39syl5bir 218 . . . . . . . . . . . . . . . . . 18  |-  ( ( ch  /\  E. f  e.  w  th )  ->  ( ( g  Fn  n  /\  ph'  /\  ps' )  -> 
g  e.  w ) )
4140ex 434 . . . . . . . . . . . . . . . . 17  |-  ( ch 
->  ( E. f  e.  w  th  ->  (
( g  Fn  n  /\  ph'  /\  ps' )  ->  g  e.  w ) ) )
4218, 41embantd 54 . . . . . . . . . . . . . . . 16  |-  ( ch 
->  ( ( ch  ->  E. f  e.  w  th )  ->  ( ( g  Fn  n  /\  ph'  /\  ps' )  -> 
g  e.  w ) ) )
4342impd 431 . . . . . . . . . . . . . . 15  |-  ( ch 
->  ( ( ( ch 
->  E. f  e.  w  th )  /\  (
g  Fn  n  /\  ph' 
/\  ps' ) )  -> 
g  e.  w ) )
4417, 43sylbir 213 . . . . . . . . . . . . . 14  |-  ( ( ta  /\  n  e.  D )  ->  (
( ( ch  ->  E. f  e.  w  th )  /\  ( g  Fn  n  /\  ph'  /\  ps' ) )  ->  g  e.  w
) )
4544expimpd 603 . . . . . . . . . . . . 13  |-  ( ta 
->  ( ( n  e.  D  /\  ( ( ch  ->  E. f  e.  w  th )  /\  ( g  Fn  n  /\  ph'  /\  ps' ) ) )  ->  g  e.  w
) )
4614, 45syl5bi 217 . . . . . . . . . . . 12  |-  ( ta 
->  ( ( ( ch 
->  E. f  e.  w  th )  /\  (
n  e.  D  /\  ( g  Fn  n  /\  ph'  /\  ps' ) ) )  ->  g  e.  w
) )
4746exlimdv 1700 . . . . . . . . . . 11  |-  ( ta 
->  ( E. n ( ( ch  ->  E. f  e.  w  th )  /\  ( n  e.  D  /\  ( g  Fn  n  /\  ph'  /\  ps' ) ) )  ->  g  e.  w
) )
4813, 47syl5 32 . . . . . . . . . 10  |-  ( ta 
->  ( ( A. n
( ch  ->  E. f  e.  w  th )  /\  E. n ( n  e.  D  /\  (
g  Fn  n  /\  ph' 
/\  ps' ) ) )  ->  g  e.  w
) )
4948expdimp 437 . . . . . . . . 9  |-  ( ( ta  /\  A. n
( ch  ->  E. f  e.  w  th )
)  ->  ( E. n ( n  e.  D  /\  ( g  Fn  n  /\  ph'  /\  ps' ) )  ->  g  e.  w
) )
5012, 49syl5bi 217 . . . . . . . 8  |-  ( ( ta  /\  A. n
( ch  ->  E. f  e.  w  th )
)  ->  ( E. n  e.  D  (
g  Fn  n  /\  ph' 
/\  ps' )  ->  g  e.  w ) )
5150abssdv 3574 . . . . . . 7  |-  ( ( ta  /\  A. n
( ch  ->  E. f  e.  w  th )
)  ->  { g  |  E. n  e.  D  ( g  Fn  n  /\  ph'  /\  ps' ) }  C_  w )
5211, 51syl5eqss 3548 . . . . . 6  |-  ( ( ta  /\  A. n
( ch  ->  E. f  e.  w  th )
)  ->  B  C_  w
)
53 vex 3116 . . . . . . 7  |-  w  e. 
_V
5453ssex 4591 . . . . . 6  |-  ( B 
C_  w  ->  B  e.  _V )
5552, 54syl 16 . . . . 5  |-  ( ( ta  /\  A. n
( ch  ->  E. f  e.  w  th )
)  ->  B  e.  _V )
5655ex 434 . . . 4  |-  ( ta 
->  ( A. n ( ch  ->  E. f  e.  w  th )  ->  B  e.  _V )
)
5756exlimdv 1700 . . 3  |-  ( ta 
->  ( E. w A. n ( ch  ->  E. f  e.  w  th )  ->  B  e.  _V ) )
587, 57mpi 17 . 2  |-  ( ta 
->  B  e.  _V )
591, 58sylbir 213 1  |-  ( ( R  FrSe  A  /\  X  e.  A )  ->  B  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973   A.wal 1377    = wceq 1379   E.wex 1596    e. wcel 1767   E!weu 2275   {cab 2452   A.wral 2814   E.wrex 2815   _Vcvv 3113   [.wsbc 3331    \ cdif 3473    C_ wss 3476   (/)c0 3785   {csn 4027   U_ciun 4325   suc csuc 4880    Fn wfn 5583   ` cfv 5588   omcom 6684    predc-bnj14 32838    FrSe w-bnj15 32842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-reg 8018  ax-inf2 8058
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-om 6685  df-1o 7130  df-bnj17 32837  df-bnj14 32839  df-bnj13 32841  df-bnj15 32843
This theorem is referenced by:  bnj893  33083
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