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Theorem bnj149 28907
Description: Technical lemma for bnj151 28909. 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
bnj149.1  |-  ( th1  <->  (
( R  FrSe  A  /\  x  e.  A
)  ->  E* f
( f  Fn  1o  /\  ph'  /\  ps' ) ) )
bnj149.2  |-  ( ze0  <->  (
f  Fn  1o  /\  ph' 
/\  ps' ) )
bnj149.3  |-  ( ze1  <->  [. g  /  f ]. ze0 )
bnj149.4  |-  ( ph1  <->  [. g  /  f ]. ph' )
bnj149.5  |-  ( ps1  <->  [. g  /  f ]. ps' )
bnj149.6  |-  ( ph'  <->  (
f `  (/) )  = 
pred ( x ,  A ,  R ) )
Assertion
Ref Expression
bnj149  |-  th1
Distinct variable groups:    A, f,
g, x    R, f,
g, x    f, ze1    g, ze0
Allowed substitution hints:    ph'( x, f, g)    ps'( x, f, g)    ze0( x, f)    ph1( x, f, g)    ps1( x, f, g)    th1( x, f, g)    ze1( x, g)

Proof of Theorem bnj149
StepHypRef Expression
1 simpr1 961 . . . . . . . 8  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  ( f  Fn  1o  /\  ph'  /\  ps' ) )  ->  f  Fn  1o )
2 df1o2 6491 . . . . . . . . 9  |-  1o  =  { (/) }
32fneq2i 5339 . . . . . . . 8  |-  ( f  Fn  1o  <->  f  Fn  {
(/) } )
41, 3sylib 188 . . . . . . 7  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  ( f  Fn  1o  /\  ph'  /\  ps' ) )  ->  f  Fn  { (/)
} )
5 simpr2 962 . . . . . . . . . 10  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  ( f  Fn  1o  /\  ph'  /\  ps' ) )  ->  ph' )
6 bnj149.6 . . . . . . . . . 10  |-  ( ph'  <->  (
f `  (/) )  = 
pred ( x ,  A ,  R ) )
75, 6sylib 188 . . . . . . . . 9  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  ( f  Fn  1o  /\  ph'  /\  ps' ) )  ->  ( f `  (/) )  =  pred (
x ,  A ,  R ) )
8 fvex 5539 . . . . . . . . . 10  |-  ( f `
 (/) )  e.  _V
98elsnc 3663 . . . . . . . . 9  |-  ( ( f `  (/) )  e. 
{  pred ( x ,  A ,  R ) }  <->  ( f `  (/) )  =  pred (
x ,  A ,  R ) )
107, 9sylibr 203 . . . . . . . 8  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  ( f  Fn  1o  /\  ph'  /\  ps' ) )  ->  ( f `  (/) )  e.  {  pred ( x ,  A ,  R ) } )
11 0ex 4150 . . . . . . . . 9  |-  (/)  e.  _V
12 fveq2 5525 . . . . . . . . . 10  |-  ( g  =  (/)  ->  ( f `
 g )  =  ( f `  (/) ) )
1312eleq1d 2349 . . . . . . . . 9  |-  ( g  =  (/)  ->  ( ( f `  g )  e.  {  pred (
x ,  A ,  R ) }  <->  ( f `  (/) )  e.  {  pred ( x ,  A ,  R ) } ) )
1411, 13ralsn 3674 . . . . . . . 8  |-  ( A. g  e.  { (/) }  (
f `  g )  e.  {  pred ( x ,  A ,  R ) }  <->  ( f `  (/) )  e.  {  pred ( x ,  A ,  R ) } )
1510, 14sylibr 203 . . . . . . 7  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  ( f  Fn  1o  /\  ph'  /\  ps' ) )  ->  A. g  e.  { (/)
}  ( f `  g )  e.  {  pred ( x ,  A ,  R ) } )
16 ffnfv 5685 . . . . . . 7  |-  ( f : { (/) } --> {  pred ( x ,  A ,  R ) }  <->  ( f  Fn  { (/) }  /\  A. g  e.  { (/) }  (
f `  g )  e.  {  pred ( x ,  A ,  R ) } ) )
174, 15, 16sylanbrc 645 . . . . . 6  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  ( f  Fn  1o  /\  ph'  /\  ps' ) )  ->  f : { (/)
} --> {  pred (
x ,  A ,  R ) } )
18 bnj93 28895 . . . . . . . 8  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  pred ( x ,  A ,  R )  e.  _V )
1918adantr 451 . . . . . . 7  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  ( f  Fn  1o  /\  ph'  /\  ps' ) )  ->  pred ( x ,  A ,  R )  e.  _V )
20 fsng 5697 . . . . . . 7  |-  ( (
(/)  e.  _V  /\  pred ( x ,  A ,  R )  e.  _V )  ->  ( f : { (/) } --> {  pred ( x ,  A ,  R ) }  <->  f  =  { <. (/) ,  pred (
x ,  A ,  R ) >. } ) )
2111, 19, 20sylancr 644 . . . . . 6  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  ( f  Fn  1o  /\  ph'  /\  ps' ) )  ->  ( f : { (/) } --> {  pred ( x ,  A ,  R ) }  <->  f  =  { <. (/) ,  pred (
x ,  A ,  R ) >. } ) )
2217, 21mpbid 201 . . . . 5  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  ( f  Fn  1o  /\  ph'  /\  ps' ) )  ->  f  =  { <.
(/) ,  pred ( x ,  A ,  R
) >. } )
2322ex 423 . . . 4  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  ( ( f  Fn  1o  /\  ph'  /\  ps' )  -> 
f  =  { <. (/)
,  pred ( x ,  A ,  R )
>. } ) )
2423alrimiv 1617 . . 3  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  A. f ( ( f  Fn  1o  /\  ph' 
/\  ps' )  ->  f  =  { <. (/) ,  pred (
x ,  A ,  R ) >. } ) )
25 mo2icl 2944 . . 3  |-  ( A. f ( ( f  Fn  1o  /\  ph'  /\  ps' )  -> 
f  =  { <. (/)
,  pred ( x ,  A ,  R )
>. } )  ->  E* f ( f  Fn  1o  /\  ph'  /\  ps' ) )
2624, 25syl 15 . 2  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  E* f ( f  Fn  1o  /\  ph'  /\  ps' ) )
27 bnj149.1 . 2  |-  ( th1  <->  (
( R  FrSe  A  /\  x  e.  A
)  ->  E* f
( f  Fn  1o  /\  ph'  /\  ps' ) ) )
2826, 27mpbir 200 1  |-  th1
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   A.wal 1527    = wceq 1623    e. wcel 1684   E*wmo 2144   A.wral 2543   _Vcvv 2788   [.wsbc 2991   (/)c0 3455   {csn 3640   <.cop 3643    Fn wfn 5250   -->wf 5251   ` cfv 5255   1oc1o 6472    predc-bnj14 28713    FrSe w-bnj15 28717
This theorem is referenced by:  bnj151  28909
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1o 6479  df-bnj13 28716  df-bnj15 28718
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