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Theorem bnj149 29474
Description: Technical lemma for bnj151 29476. 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 1011 . . . . . . . 8  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  ( f  Fn  1o  /\  ph'  /\  ps' ) )  ->  f  Fn  1o )
2 df1o2 7202 . . . . . . . . 9  |-  1o  =  { (/) }
32fneq2i 5689 . . . . . . . 8  |-  ( f  Fn  1o  <->  f  Fn  {
(/) } )
41, 3sylib 199 . . . . . . 7  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  ( f  Fn  1o  /\  ph'  /\  ps' ) )  ->  f  Fn  { (/)
} )
5 simpr2 1012 . . . . . . . . . 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 199 . . . . . . . . 9  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  ( f  Fn  1o  /\  ph'  /\  ps' ) )  ->  ( f `  (/) )  =  pred (
x ,  A ,  R ) )
8 fvex 5891 . . . . . . . . . 10  |-  ( f `
 (/) )  e.  _V
98elsnc 4026 . . . . . . . . 9  |-  ( ( f `  (/) )  e. 
{  pred ( x ,  A ,  R ) }  <->  ( f `  (/) )  =  pred (
x ,  A ,  R ) )
107, 9sylibr 215 . . . . . . . 8  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  ( f  Fn  1o  /\  ph'  /\  ps' ) )  ->  ( f `  (/) )  e.  {  pred ( x ,  A ,  R ) } )
11 0ex 4557 . . . . . . . . 9  |-  (/)  e.  _V
12 fveq2 5881 . . . . . . . . . 10  |-  ( g  =  (/)  ->  ( f `
 g )  =  ( f `  (/) ) )
1312eleq1d 2498 . . . . . . . . 9  |-  ( g  =  (/)  ->  ( ( f `  g )  e.  {  pred (
x ,  A ,  R ) }  <->  ( f `  (/) )  e.  {  pred ( x ,  A ,  R ) } ) )
1411, 13ralsn 4041 . . . . . . . 8  |-  ( A. g  e.  { (/) }  (
f `  g )  e.  {  pred ( x ,  A ,  R ) }  <->  ( f `  (/) )  e.  {  pred ( x ,  A ,  R ) } )
1510, 14sylibr 215 . . . . . . 7  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  ( f  Fn  1o  /\  ph'  /\  ps' ) )  ->  A. g  e.  { (/)
}  ( f `  g )  e.  {  pred ( x ,  A ,  R ) } )
16 ffnfv 6064 . . . . . . 7  |-  ( f : { (/) } --> {  pred ( x ,  A ,  R ) }  <->  ( f  Fn  { (/) }  /\  A. g  e.  { (/) }  (
f `  g )  e.  {  pred ( x ,  A ,  R ) } ) )
174, 15, 16sylanbrc 668 . . . . . 6  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  ( f  Fn  1o  /\  ph'  /\  ps' ) )  ->  f : { (/)
} --> {  pred (
x ,  A ,  R ) } )
18 bnj93 29462 . . . . . . . 8  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  pred ( x ,  A ,  R )  e.  _V )
1918adantr 466 . . . . . . 7  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  ( f  Fn  1o  /\  ph'  /\  ps' ) )  ->  pred ( x ,  A ,  R )  e.  _V )
20 fsng 6078 . . . . . . 7  |-  ( (
(/)  e.  _V  /\  pred ( x ,  A ,  R )  e.  _V )  ->  ( f : { (/) } --> {  pred ( x ,  A ,  R ) }  <->  f  =  { <. (/) ,  pred (
x ,  A ,  R ) >. } ) )
2111, 19, 20sylancr 667 . . . . . 6  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  ( f  Fn  1o  /\  ph'  /\  ps' ) )  ->  ( f : { (/) } --> {  pred ( x ,  A ,  R ) }  <->  f  =  { <. (/) ,  pred (
x ,  A ,  R ) >. } ) )
2217, 21mpbid 213 . . . . 5  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  ( f  Fn  1o  /\  ph'  /\  ps' ) )  ->  f  =  { <.
(/) ,  pred ( x ,  A ,  R
) >. } )
2322ex 435 . . . 4  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  ( ( f  Fn  1o  /\  ph'  /\  ps' )  -> 
f  =  { <. (/)
,  pred ( x ,  A ,  R )
>. } ) )
2423alrimiv 1766 . . 3  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  A. f ( ( f  Fn  1o  /\  ph' 
/\  ps' )  ->  f  =  { <. (/) ,  pred (
x ,  A ,  R ) >. } ) )
25 mo2icl 3256 . . 3  |-  ( A. f ( ( f  Fn  1o  /\  ph'  /\  ps' )  -> 
f  =  { <. (/)
,  pred ( x ,  A ,  R )
>. } )  ->  E* f ( f  Fn  1o  /\  ph'  /\  ps' ) )
2624, 25syl 17 . 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 212 1  |-  th1
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982   A.wal 1435    = wceq 1437    e. wcel 1870   E*wmo 2267   A.wral 2782   _Vcvv 3087   [.wsbc 3305   (/)c0 3767   {csn 4002   <.cop 4008    Fn wfn 5596   -->wf 5597   ` cfv 5601   1oc1o 7183    predc-bnj14 29281    FrSe w-bnj15 29285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-1o 7190  df-bnj13 29284  df-bnj15 29286
This theorem is referenced by:  bnj151  29476
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