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Theorem bnj150 32202
Description: Technical lemma for bnj151 32203. 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
bnj150.1  |-  ( ph  <->  ( f `  (/) )  = 
pred ( x ,  A ,  R ) )
bnj150.2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj150.3  |-  ( ze  <->  ( ( R  FrSe  A  /\  x  e.  A
)  ->  ( f  Fn  n  /\  ph  /\  ps ) ) )
bnj150.4  |-  ( ph'  <->  [. 1o  /  n ]. ph )
bnj150.5  |-  ( ps'  <->  [. 1o  /  n ]. ps )
bnj150.6  |-  ( th0  <->  (
( R  FrSe  A  /\  x  e.  A
)  ->  E. f
( f  Fn  1o  /\  ph'  /\  ps' ) ) )
bnj150.7  |-  ( ze'  <->  [. 1o  /  n ]. ze )
bnj150.8  |-  F  =  { <. (/) ,  pred (
x ,  A ,  R ) >. }
bnj150.9  |-  ( ph"  <->  [. F  / 
f ]. ph' )
bnj150.10  |-  ( ps"  <->  [. F  / 
f ]. ps' )
bnj150.11  |-  ( ze"  <->  [. F  / 
f ]. ze' )
Assertion
Ref Expression
bnj150  |-  th0
Distinct variable groups:    A, f, n, x    f, F, i, y    R, f, n, x   
f, ze"    i, n, y
Allowed substitution hints:    ph( x, y, f, i, n)    ps( x, y, f, i, n)    ze( x, y, f, i, n)    A( y, i)    R( y, i)    F( x, n)    ph'( x, y, f, i, n)    ps'( x, y, f, i, n)    ze'( x, y, f, i, n)    ph"( x, y, f, i, n)    ps"( x, y, f, i, n)    ze"( x, y, i, n)    th0( x, y, f, i, n)

Proof of Theorem bnj150
StepHypRef Expression
1 0ex 4531 . . . . . . . . . 10  |-  (/)  e.  _V
2 bnj93 32189 . . . . . . . . . 10  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  pred ( x ,  A ,  R )  e.  _V )
3 funsng 5573 . . . . . . . . . 10  |-  ( (
(/)  e.  _V  /\  pred ( x ,  A ,  R )  e.  _V )  ->  Fun  { <. (/) ,  pred ( x ,  A ,  R ) >. } )
41, 2, 3sylancr 663 . . . . . . . . 9  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  Fun  { <. (/) ,  pred ( x ,  A ,  R ) >. } )
5 bnj150.8 . . . . . . . . . 10  |-  F  =  { <. (/) ,  pred (
x ,  A ,  R ) >. }
65funeqi 5547 . . . . . . . . 9  |-  ( Fun 
F  <->  Fun  { <. (/) ,  pred ( x ,  A ,  R ) >. } )
74, 6sylibr 212 . . . . . . . 8  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  Fun  F )
85bnj96 32191 . . . . . . . 8  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  dom  F  =  1o )
97, 8bnj1422 32164 . . . . . . 7  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  F  Fn  1o )
105bnj97 32192 . . . . . . . 8  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  ( F `  (/) )  = 
pred ( x ,  A ,  R ) )
11 bnj150.1 . . . . . . . . 9  |-  ( ph  <->  ( f `  (/) )  = 
pred ( x ,  A ,  R ) )
12 bnj150.4 . . . . . . . . 9  |-  ( ph'  <->  [. 1o  /  n ]. ph )
13 bnj150.9 . . . . . . . . 9  |-  ( ph"  <->  [. F  / 
f ]. ph' )
1411, 12, 13, 5bnj125 32198 . . . . . . . 8  |-  ( ph"  <->  ( F `  (/) )  =  pred ( x ,  A ,  R ) )
1510, 14sylibr 212 . . . . . . 7  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  ph" )
169, 15jca 532 . . . . . 6  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  ( F  Fn  1o  /\  ph" ) )
17 bnj98 32193 . . . . . . 7  |-  A. i  e.  om  ( suc  i  e.  1o  ->  ( F `  suc  i )  = 
U_ y  e.  ( F `  i ) 
pred ( y ,  A ,  R ) )
18 bnj150.2 . . . . . . . 8  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
19 bnj150.5 . . . . . . . 8  |-  ( ps'  <->  [. 1o  /  n ]. ps )
20 bnj150.10 . . . . . . . 8  |-  ( ps"  <->  [. F  / 
f ]. ps' )
2118, 19, 20, 5bnj126 32199 . . . . . . 7  |-  ( ps"  <->  A. i  e.  om  ( suc  i  e.  1o  ->  ( F `  suc  i )  = 
U_ y  e.  ( F `  i ) 
pred ( y ,  A ,  R ) ) )
2217, 21mpbir 209 . . . . . 6  |-  ps"
2316, 22jctir 538 . . . . 5  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  ( ( F  Fn  1o  /\  ph" )  /\  ps" ) )
24 df-3an 967 . . . . 5  |-  ( ( F  Fn  1o  /\  ph"  /\  ps" )  <->  ( ( F  Fn  1o  /\  ph" )  /\  ps" ) )
2523, 24sylibr 212 . . . 4  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  ( F  Fn  1o  /\  ph" 
/\  ps" ) )
26 bnj150.11 . . . . 5  |-  ( ze"  <->  [. F  / 
f ]. ze' )
27 bnj150.3 . . . . . 6  |-  ( ze  <->  ( ( R  FrSe  A  /\  x  e.  A
)  ->  ( f  Fn  n  /\  ph  /\  ps ) ) )
28 bnj150.7 . . . . . 6  |-  ( ze'  <->  [. 1o  /  n ]. ze )
2927, 28, 12, 19bnj121 32196 . . . . 5  |-  ( ze'  <->  (
( R  FrSe  A  /\  x  e.  A
)  ->  ( f  Fn  1o  /\  ph'  /\  ps' ) ) )
305, 13, 20, 26, 29bnj124 32197 . . . 4  |-  ( ze"  <->  ( ( R  FrSe  A  /\  x  e.  A )  ->  ( F  Fn  1o  /\  ph"  /\  ps" ) ) )
3125, 30mpbir 209 . . 3  |-  ze"
325bnj95 32190 . . . 4  |-  F  e. 
_V
33 sbceq1a 3305 . . . . 5  |-  ( f  =  F  ->  ( ze'  <->  [. F  /  f ]. ze' ) )
3433, 26syl6bbr 263 . . . 4  |-  ( f  =  F  ->  ( ze'  <->  ze" ) )
3532, 34spcev 3170 . . 3  |-  ( ze"  ->  E. f ze' )
3631, 35ax-mp 5 . 2  |-  E. f ze'
37 bnj150.6 . . . 4  |-  ( th0  <->  (
( R  FrSe  A  /\  x  e.  A
)  ->  E. f
( f  Fn  1o  /\  ph'  /\  ps' ) ) )
38 19.37v 1930 . . . 4  |-  ( E. f ( ( R 
FrSe  A  /\  x  e.  A )  ->  (
f  Fn  1o  /\  ph' 
/\  ps' ) )  <->  ( ( R  FrSe  A  /\  x  e.  A )  ->  E. f
( f  Fn  1o  /\  ph'  /\  ps' ) ) )
3937, 38bitr4i 252 . . 3  |-  ( th0  <->  E. f ( ( R 
FrSe  A  /\  x  e.  A )  ->  (
f  Fn  1o  /\  ph' 
/\  ps' ) ) )
4039, 29bnj133 32049 . 2  |-  ( th0  <->  E. f ze' )
4136, 40mpbir 209 1  |-  th0
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370   E.wex 1587    e. wcel 1758   A.wral 2799   _Vcvv 3078   [.wsbc 3294   (/)c0 3746   {csn 3986   <.cop 3992   U_ciun 4280   suc csuc 4830   Fun wfun 5521    Fn wfn 5522   ` cfv 5527   omcom 6587   1oc1o 7024    predc-bnj14 32009    FrSe w-bnj15 32013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-id 4745  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-iota 5490  df-fun 5529  df-fn 5530  df-fv 5535  df-1o 7031  df-bnj13 32012  df-bnj15 32014
This theorem is referenced by:  bnj151  32203
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