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Theorem fidcenumlemrks 6660
Description: Lemma for fidcenum 6663. Induction step for fidcenumlemrk 6661. (Contributed by Jim Kingdon, 20-Oct-2022.)
Hypotheses
Ref Expression
fidcenumlemr.dc  |-  ( ph  ->  A. x  e.  A  A. y  e.  A DECID  x  =  y )
fidcenumlemr.n  |-  ( ph  ->  N  e.  om )
fidcenumlemr.f  |-  ( ph  ->  F : N -onto-> A
)
fidcenumlemrks.j  |-  ( ph  ->  J  e.  om )
fidcenumlemrks.jn  |-  ( ph  ->  suc  J  C_  N
)
fidcenumlemrks.h  |-  ( ph  ->  ( X  e.  ( F " J )  \/  -.  X  e.  ( F " J
) ) )
fidcenumlemrks.x  |-  ( ph  ->  X  e.  A )
Assertion
Ref Expression
fidcenumlemrks  |-  ( ph  ->  ( X  e.  ( F " suc  J
)  \/  -.  X  e.  ( F " suc  J ) ) )
Distinct variable groups:    x, A, y   
y, F    y, J    x, X, y
Allowed substitution hints:    ph( x, y)    F( x)    J( x)    N( x, y)

Proof of Theorem fidcenumlemrks
StepHypRef Expression
1 simpr 108 . . . . 5  |-  ( (
ph  /\  X  e.  ( F " J ) )  ->  X  e.  ( F " J ) )
2 elun1 3167 . . . . 5  |-  ( X  e.  ( F " J )  ->  X  e.  ( ( F " J )  u.  ( F " { J }
) ) )
31, 2syl 14 . . . 4  |-  ( (
ph  /\  X  e.  ( F " J ) )  ->  X  e.  ( ( F " J )  u.  ( F " { J }
) ) )
4 df-suc 4198 . . . . . . 7  |-  suc  J  =  ( J  u.  { J } )
54imaeq2i 4772 . . . . . 6  |-  ( F
" suc  J )  =  ( F "
( J  u.  { J } ) )
6 imaundi 4844 . . . . . 6  |-  ( F
" ( J  u.  { J } ) )  =  ( ( F
" J )  u.  ( F " { J } ) )
75, 6eqtri 2108 . . . . 5  |-  ( F
" suc  J )  =  ( ( F
" J )  u.  ( F " { J } ) )
87eleq2i 2154 . . . 4  |-  ( X  e.  ( F " suc  J )  <->  X  e.  ( ( F " J )  u.  ( F " { J }
) ) )
93, 8sylibr 132 . . 3  |-  ( (
ph  /\  X  e.  ( F " J ) )  ->  X  e.  ( F " suc  J
) )
109orcd 687 . 2  |-  ( (
ph  /\  X  e.  ( F " J ) )  ->  ( X  e.  ( F " suc  J )  \/  -.  X  e.  ( F " suc  J ) ) )
11 simpr 108 . . . . . . 7  |-  ( ( ( ph  /\  -.  X  e.  ( F " J ) )  /\  X  =  ( F `  J ) )  ->  X  =  ( F `  J ) )
12 fidcenumlemrks.x . . . . . . . . . 10  |-  ( ph  ->  X  e.  A )
13 elsng 3461 . . . . . . . . . 10  |-  ( X  e.  A  ->  ( X  e.  { ( F `  J ) } 
<->  X  =  ( F `
 J ) ) )
1412, 13syl 14 . . . . . . . . 9  |-  ( ph  ->  ( X  e.  {
( F `  J
) }  <->  X  =  ( F `  J ) ) )
15 fidcenumlemr.f . . . . . . . . . . . 12  |-  ( ph  ->  F : N -onto-> A
)
16 fofn 5235 . . . . . . . . . . . 12  |-  ( F : N -onto-> A  ->  F  Fn  N )
1715, 16syl 14 . . . . . . . . . . 11  |-  ( ph  ->  F  Fn  N )
18 fidcenumlemrks.jn . . . . . . . . . . . 12  |-  ( ph  ->  suc  J  C_  N
)
19 fidcenumlemrks.j . . . . . . . . . . . . 13  |-  ( ph  ->  J  e.  om )
20 sucidg 4243 . . . . . . . . . . . . 13  |-  ( J  e.  om  ->  J  e.  suc  J )
2119, 20syl 14 . . . . . . . . . . . 12  |-  ( ph  ->  J  e.  suc  J
)
2218, 21sseldd 3026 . . . . . . . . . . 11  |-  ( ph  ->  J  e.  N )
23 fnsnfv 5363 . . . . . . . . . . 11  |-  ( ( F  Fn  N  /\  J  e.  N )  ->  { ( F `  J ) }  =  ( F " { J } ) )
2417, 22, 23syl2anc 403 . . . . . . . . . 10  |-  ( ph  ->  { ( F `  J ) }  =  ( F " { J } ) )
2524eleq2d 2157 . . . . . . . . 9  |-  ( ph  ->  ( X  e.  {
( F `  J
) }  <->  X  e.  ( F " { J } ) ) )
2614, 25bitr3d 188 . . . . . . . 8  |-  ( ph  ->  ( X  =  ( F `  J )  <-> 
X  e.  ( F
" { J }
) ) )
2726ad2antrr 472 . . . . . . 7  |-  ( ( ( ph  /\  -.  X  e.  ( F " J ) )  /\  X  =  ( F `  J ) )  -> 
( X  =  ( F `  J )  <-> 
X  e.  ( F
" { J }
) ) )
2811, 27mpbid 145 . . . . . 6  |-  ( ( ( ph  /\  -.  X  e.  ( F " J ) )  /\  X  =  ( F `  J ) )  ->  X  e.  ( F " { J } ) )
29 elun2 3168 . . . . . 6  |-  ( X  e.  ( F " { J } )  ->  X  e.  ( ( F " J )  u.  ( F " { J } ) ) )
3028, 29syl 14 . . . . 5  |-  ( ( ( ph  /\  -.  X  e.  ( F " J ) )  /\  X  =  ( F `  J ) )  ->  X  e.  ( ( F " J )  u.  ( F " { J } ) ) )
3130, 8sylibr 132 . . . 4  |-  ( ( ( ph  /\  -.  X  e.  ( F " J ) )  /\  X  =  ( F `  J ) )  ->  X  e.  ( F " suc  J ) )
3231orcd 687 . . 3  |-  ( ( ( ph  /\  -.  X  e.  ( F " J ) )  /\  X  =  ( F `  J ) )  -> 
( X  e.  ( F " suc  J
)  \/  -.  X  e.  ( F " suc  J ) ) )
33 simplr 497 . . . . . . 7  |-  ( ( ( ph  /\  -.  X  e.  ( F " J ) )  /\  -.  X  =  ( F `  J )
)  ->  -.  X  e.  ( F " J
) )
34 simpr 108 . . . . . . . 8  |-  ( ( ( ph  /\  -.  X  e.  ( F " J ) )  /\  -.  X  =  ( F `  J )
)  ->  -.  X  =  ( F `  J ) )
3526ad2antrr 472 . . . . . . . 8  |-  ( ( ( ph  /\  -.  X  e.  ( F " J ) )  /\  -.  X  =  ( F `  J )
)  ->  ( X  =  ( F `  J )  <->  X  e.  ( F " { J } ) ) )
3634, 35mtbid 632 . . . . . . 7  |-  ( ( ( ph  /\  -.  X  e.  ( F " J ) )  /\  -.  X  =  ( F `  J )
)  ->  -.  X  e.  ( F " { J } ) )
37 ioran 704 . . . . . . 7  |-  ( -.  ( X  e.  ( F " J )  \/  X  e.  ( F " { J } ) )  <->  ( -.  X  e.  ( F " J )  /\  -.  X  e.  ( F " { J } ) ) )
3833, 36, 37sylanbrc 408 . . . . . 6  |-  ( ( ( ph  /\  -.  X  e.  ( F " J ) )  /\  -.  X  =  ( F `  J )
)  ->  -.  ( X  e.  ( F " J )  \/  X  e.  ( F " { J } ) ) )
39 elun 3141 . . . . . 6  |-  ( X  e.  ( ( F
" J )  u.  ( F " { J } ) )  <->  ( X  e.  ( F " J
)  \/  X  e.  ( F " { J } ) ) )
4038, 39sylnibr 637 . . . . 5  |-  ( ( ( ph  /\  -.  X  e.  ( F " J ) )  /\  -.  X  =  ( F `  J )
)  ->  -.  X  e.  ( ( F " J )  u.  ( F " { J }
) ) )
4140, 8sylnibr 637 . . . 4  |-  ( ( ( ph  /\  -.  X  e.  ( F " J ) )  /\  -.  X  =  ( F `  J )
)  ->  -.  X  e.  ( F " suc  J ) )
4241olcd 688 . . 3  |-  ( ( ( ph  /\  -.  X  e.  ( F " J ) )  /\  -.  X  =  ( F `  J )
)  ->  ( X  e.  ( F " suc  J )  \/  -.  X  e.  ( F " suc  J ) ) )
43 fof 5233 . . . . . . . 8  |-  ( F : N -onto-> A  ->  F : N --> A )
4415, 43syl 14 . . . . . . 7  |-  ( ph  ->  F : N --> A )
4544, 22ffvelrnd 5435 . . . . . 6  |-  ( ph  ->  ( F `  J
)  e.  A )
46 fidcenumlemr.dc . . . . . 6  |-  ( ph  ->  A. x  e.  A  A. y  e.  A DECID  x  =  y )
47 eqeq1 2094 . . . . . . . 8  |-  ( x  =  X  ->  (
x  =  y  <->  X  =  y ) )
4847dcbid 786 . . . . . . 7  |-  ( x  =  X  ->  (DECID  x  =  y  <-> DECID  X  =  y )
)
49 eqeq2 2097 . . . . . . . 8  |-  ( y  =  ( F `  J )  ->  ( X  =  y  <->  X  =  ( F `  J ) ) )
5049dcbid 786 . . . . . . 7  |-  ( y  =  ( F `  J )  ->  (DECID  X  =  y  <-> DECID  X  =  ( F `  J ) ) )
5148, 50rspc2va 2735 . . . . . 6  |-  ( ( ( X  e.  A  /\  ( F `  J
)  e.  A )  /\  A. x  e.  A  A. y  e.  A DECID  x  =  y )  -> DECID 
X  =  ( F `
 J ) )
5212, 45, 46, 51syl21anc 1173 . . . . 5  |-  ( ph  -> DECID  X  =  ( F `  J ) )
53 exmiddc 782 . . . . 5  |-  (DECID  X  =  ( F `  J
)  ->  ( X  =  ( F `  J )  \/  -.  X  =  ( F `  J ) ) )
5452, 53syl 14 . . . 4  |-  ( ph  ->  ( X  =  ( F `  J )  \/  -.  X  =  ( F `  J
) ) )
5554adantr 270 . . 3  |-  ( (
ph  /\  -.  X  e.  ( F " J
) )  ->  ( X  =  ( F `  J )  \/  -.  X  =  ( F `  J ) ) )
5632, 42, 55mpjaodan 747 . 2  |-  ( (
ph  /\  -.  X  e.  ( F " J
) )  ->  ( X  e.  ( F " suc  J )  \/ 
-.  X  e.  ( F " suc  J
) ) )
57 fidcenumlemrks.h . 2  |-  ( ph  ->  ( X  e.  ( F " J )  \/  -.  X  e.  ( F " J
) ) )
5810, 56, 57mpjaodan 747 1  |-  ( ph  ->  ( X  e.  ( F " suc  J
)  \/  -.  X  e.  ( F " suc  J ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 664  DECID wdc 780    = wceq 1289    e. wcel 1438   A.wral 2359    u. cun 2997    C_ wss 2999   {csn 3446   suc csuc 4192   omcom 4405   "cima 4441    Fn wfn 5010   -->wf 5011   -onto->wfo 5013   ` cfv 5015
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-id 4120  df-suc 4198  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-fo 5021  df-fv 5023
This theorem is referenced by:  fidcenumlemrk  6661
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