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Theorem bnj149 29173
Description: Technical lemma for bnj151 29175. 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 963 . . . . . . . 8  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  ( f  Fn  1o  /\  ph'  /\  ps' ) )  ->  f  Fn  1o )
2 df1o2 6728 . . . . . . . . 9  |-  1o  =  { (/) }
32fneq2i 5532 . . . . . . . 8  |-  ( f  Fn  1o  <->  f  Fn  {
(/) } )
41, 3sylib 189 . . . . . . 7  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  ( f  Fn  1o  /\  ph'  /\  ps' ) )  ->  f  Fn  { (/)
} )
5 simpr2 964 . . . . . . . . . 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 189 . . . . . . . . 9  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  ( f  Fn  1o  /\  ph'  /\  ps' ) )  ->  ( f `  (/) )  =  pred (
x ,  A ,  R ) )
8 fvex 5734 . . . . . . . . . 10  |-  ( f `
 (/) )  e.  _V
98elsnc 3829 . . . . . . . . 9  |-  ( ( f `  (/) )  e. 
{  pred ( x ,  A ,  R ) }  <->  ( f `  (/) )  =  pred (
x ,  A ,  R ) )
107, 9sylibr 204 . . . . . . . 8  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  ( f  Fn  1o  /\  ph'  /\  ps' ) )  ->  ( f `  (/) )  e.  {  pred ( x ,  A ,  R ) } )
11 0ex 4331 . . . . . . . . 9  |-  (/)  e.  _V
12 fveq2 5720 . . . . . . . . . 10  |-  ( g  =  (/)  ->  ( f `
 g )  =  ( f `  (/) ) )
1312eleq1d 2501 . . . . . . . . 9  |-  ( g  =  (/)  ->  ( ( f `  g )  e.  {  pred (
x ,  A ,  R ) }  <->  ( f `  (/) )  e.  {  pred ( x ,  A ,  R ) } ) )
1411, 13ralsn 3841 . . . . . . . 8  |-  ( A. g  e.  { (/) }  (
f `  g )  e.  {  pred ( x ,  A ,  R ) }  <->  ( f `  (/) )  e.  {  pred ( x ,  A ,  R ) } )
1510, 14sylibr 204 . . . . . . 7  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  ( f  Fn  1o  /\  ph'  /\  ps' ) )  ->  A. g  e.  { (/)
}  ( f `  g )  e.  {  pred ( x ,  A ,  R ) } )
16 ffnfv 5886 . . . . . . 7  |-  ( f : { (/) } --> {  pred ( x ,  A ,  R ) }  <->  ( f  Fn  { (/) }  /\  A. g  e.  { (/) }  (
f `  g )  e.  {  pred ( x ,  A ,  R ) } ) )
174, 15, 16sylanbrc 646 . . . . . 6  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  ( f  Fn  1o  /\  ph'  /\  ps' ) )  ->  f : { (/)
} --> {  pred (
x ,  A ,  R ) } )
18 bnj93 29161 . . . . . . . 8  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  pred ( x ,  A ,  R )  e.  _V )
1918adantr 452 . . . . . . 7  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  ( f  Fn  1o  /\  ph'  /\  ps' ) )  ->  pred ( x ,  A ,  R )  e.  _V )
20 fsng 5899 . . . . . . 7  |-  ( (
(/)  e.  _V  /\  pred ( x ,  A ,  R )  e.  _V )  ->  ( f : { (/) } --> {  pred ( x ,  A ,  R ) }  <->  f  =  { <. (/) ,  pred (
x ,  A ,  R ) >. } ) )
2111, 19, 20sylancr 645 . . . . . 6  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  ( f  Fn  1o  /\  ph'  /\  ps' ) )  ->  ( f : { (/) } --> {  pred ( x ,  A ,  R ) }  <->  f  =  { <. (/) ,  pred (
x ,  A ,  R ) >. } ) )
2217, 21mpbid 202 . . . . 5  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  ( f  Fn  1o  /\  ph'  /\  ps' ) )  ->  f  =  { <.
(/) ,  pred ( x ,  A ,  R
) >. } )
2322ex 424 . . . 4  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  ( ( f  Fn  1o  /\  ph'  /\  ps' )  -> 
f  =  { <. (/)
,  pred ( x ,  A ,  R )
>. } ) )
2423alrimiv 1641 . . 3  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  A. f ( ( f  Fn  1o  /\  ph' 
/\  ps' )  ->  f  =  { <. (/) ,  pred (
x ,  A ,  R ) >. } ) )
25 mo2icl 3105 . . 3  |-  ( A. f ( ( f  Fn  1o  /\  ph'  /\  ps' )  -> 
f  =  { <. (/)
,  pred ( x ,  A ,  R )
>. } )  ->  E* f ( f  Fn  1o  /\  ph'  /\  ps' ) )
2624, 25syl 16 . 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 201 1  |-  th1
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   A.wal 1549    = wceq 1652    e. wcel 1725   E*wmo 2281   A.wral 2697   _Vcvv 2948   [.wsbc 3153   (/)c0 3620   {csn 3806   <.cop 3809    Fn wfn 5441   -->wf 5442   ` cfv 5446   1oc1o 6709    predc-bnj14 28979    FrSe w-bnj15 28983
This theorem is referenced by:  bnj151  29175
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  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-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-suc 4579  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  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-bnj13 28982  df-bnj15 28984
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