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Theorem fidcenumlemrks 6809
Description: Lemma for fidcenum 6812. Induction step for fidcenumlemrk 6810. (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.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 109 . . . . 5  |-  ( (
ph  /\  X  e.  ( F " J ) )  ->  X  e.  ( F " J ) )
2 elun1 3213 . . . . 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 4263 . . . . . . 7  |-  suc  J  =  ( J  u.  { J } )
54imaeq2i 4849 . . . . . 6  |-  ( F
" suc  J )  =  ( F "
( J  u.  { J } ) )
6 imaundi 4921 . . . . . 6  |-  ( F
" ( J  u.  { J } ) )  =  ( ( F
" J )  u.  ( F " { J } ) )
75, 6eqtri 2138 . . . . 5  |-  ( F
" suc  J )  =  ( ( F
" J )  u.  ( F " { J } ) )
87eleq2i 2184 . . . 4  |-  ( X  e.  ( F " suc  J )  <->  X  e.  ( ( F " J )  u.  ( F " { J }
) ) )
93, 8sylibr 133 . . 3  |-  ( (
ph  /\  X  e.  ( F " J ) )  ->  X  e.  ( F " suc  J
) )
109orcd 707 . 2  |-  ( (
ph  /\  X  e.  ( F " J ) )  ->  ( X  e.  ( F " suc  J )  \/  -.  X  e.  ( F " suc  J ) ) )
11 simpr 109 . . . . . . 7  |-  ( ( ( ph  /\  -.  X  e.  ( F " J ) )  /\  X  =  ( F `  J ) )  ->  X  =  ( F `  J ) )
12 fidcenumlemrks.x . . . . . . . . . 10  |-  ( ph  ->  X  e.  A )
13 elsng 3512 . . . . . . . . . 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 5317 . . . . . . . . . . . 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 4308 . . . . . . . . . . . . 13  |-  ( J  e.  om  ->  J  e.  suc  J )
2119, 20syl 14 . . . . . . . . . . . 12  |-  ( ph  ->  J  e.  suc  J
)
2218, 21sseldd 3068 . . . . . . . . . . 11  |-  ( ph  ->  J  e.  N )
23 fnsnfv 5448 . . . . . . . . . . 11  |-  ( ( F  Fn  N  /\  J  e.  N )  ->  { ( F `  J ) }  =  ( F " { J } ) )
2417, 22, 23syl2anc 408 . . . . . . . . . 10  |-  ( ph  ->  { ( F `  J ) }  =  ( F " { J } ) )
2524eleq2d 2187 . . . . . . . . 9  |-  ( ph  ->  ( X  e.  {
( F `  J
) }  <->  X  e.  ( F " { J } ) ) )
2614, 25bitr3d 189 . . . . . . . 8  |-  ( ph  ->  ( X  =  ( F `  J )  <-> 
X  e.  ( F
" { J }
) ) )
2726ad2antrr 479 . . . . . . 7  |-  ( ( ( ph  /\  -.  X  e.  ( F " J ) )  /\  X  =  ( F `  J ) )  -> 
( X  =  ( F `  J )  <-> 
X  e.  ( F
" { J }
) ) )
2811, 27mpbid 146 . . . . . 6  |-  ( ( ( ph  /\  -.  X  e.  ( F " J ) )  /\  X  =  ( F `  J ) )  ->  X  e.  ( F " { J } ) )
29 elun2 3214 . . . . . 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 133 . . . 4  |-  ( ( ( ph  /\  -.  X  e.  ( F " J ) )  /\  X  =  ( F `  J ) )  ->  X  e.  ( F " suc  J ) )
3231orcd 707 . . 3  |-  ( ( ( ph  /\  -.  X  e.  ( F " J ) )  /\  X  =  ( F `  J ) )  -> 
( X  e.  ( F " suc  J
)  \/  -.  X  e.  ( F " suc  J ) ) )
33 simplr 504 . . . . . . 7  |-  ( ( ( ph  /\  -.  X  e.  ( F " J ) )  /\  -.  X  =  ( F `  J )
)  ->  -.  X  e.  ( F " J
) )
34 simpr 109 . . . . . . . 8  |-  ( ( ( ph  /\  -.  X  e.  ( F " J ) )  /\  -.  X  =  ( F `  J )
)  ->  -.  X  =  ( F `  J ) )
3526ad2antrr 479 . . . . . . . 8  |-  ( ( ( ph  /\  -.  X  e.  ( F " J ) )  /\  -.  X  =  ( F `  J )
)  ->  ( X  =  ( F `  J )  <->  X  e.  ( F " { J } ) ) )
3634, 35mtbid 646 . . . . . . 7  |-  ( ( ( ph  /\  -.  X  e.  ( F " J ) )  /\  -.  X  =  ( F `  J )
)  ->  -.  X  e.  ( F " { J } ) )
37 ioran 726 . . . . . . 7  |-  ( -.  ( X  e.  ( F " J )  \/  X  e.  ( F " { J } ) )  <->  ( -.  X  e.  ( F " J )  /\  -.  X  e.  ( F " { J } ) ) )
3833, 36, 37sylanbrc 413 . . . . . 6  |-  ( ( ( ph  /\  -.  X  e.  ( F " J ) )  /\  -.  X  =  ( F `  J )
)  ->  -.  ( X  e.  ( F " J )  \/  X  e.  ( F " { J } ) ) )
39 elun 3187 . . . . . 6  |-  ( X  e.  ( ( F
" J )  u.  ( F " { J } ) )  <->  ( X  e.  ( F " J
)  \/  X  e.  ( F " { J } ) ) )
4038, 39sylnibr 651 . . . . 5  |-  ( ( ( ph  /\  -.  X  e.  ( F " J ) )  /\  -.  X  =  ( F `  J )
)  ->  -.  X  e.  ( ( F " J )  u.  ( F " { J }
) ) )
4140, 8sylnibr 651 . . . 4  |-  ( ( ( ph  /\  -.  X  e.  ( F " J ) )  /\  -.  X  =  ( F `  J )
)  ->  -.  X  e.  ( F " suc  J ) )
4241olcd 708 . . 3  |-  ( ( ( ph  /\  -.  X  e.  ( F " J ) )  /\  -.  X  =  ( F `  J )
)  ->  ( X  e.  ( F " suc  J )  \/  -.  X  e.  ( F " suc  J ) ) )
43 fof 5315 . . . . . . . 8  |-  ( F : N -onto-> A  ->  F : N --> A )
4415, 43syl 14 . . . . . . 7  |-  ( ph  ->  F : N --> A )
4544, 22ffvelrnd 5524 . . . . . 6  |-  ( ph  ->  ( F `  J
)  e.  A )
46 fidcenumlemr.dc . . . . . 6  |-  ( ph  ->  A. x  e.  A  A. y  e.  A DECID  x  =  y )
47 eqeq1 2124 . . . . . . . 8  |-  ( x  =  X  ->  (
x  =  y  <->  X  =  y ) )
4847dcbid 808 . . . . . . 7  |-  ( x  =  X  ->  (DECID  x  =  y  <-> DECID  X  =  y )
)
49 eqeq2 2127 . . . . . . . 8  |-  ( y  =  ( F `  J )  ->  ( X  =  y  <->  X  =  ( F `  J ) ) )
5049dcbid 808 . . . . . . 7  |-  ( y  =  ( F `  J )  ->  (DECID  X  =  y  <-> DECID  X  =  ( F `  J ) ) )
5148, 50rspc2va 2777 . . . . . 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 1200 . . . . 5  |-  ( ph  -> DECID  X  =  ( F `  J ) )
53 exmiddc 806 . . . . 5  |-  (DECID  X  =  ( F `  J
)  ->  ( X  =  ( F `  J )  \/  -.  X  =  ( F `  J ) ) )
5452, 53syl 14 . . . 4  |-  ( ph  ->  ( X  =  ( F `  J )  \/  -.  X  =  ( F `  J
) ) )
5554adantr 274 . . 3  |-  ( (
ph  /\  -.  X  e.  ( F " J
) )  ->  ( X  =  ( F `  J )  \/  -.  X  =  ( F `  J ) ) )
5632, 42, 55mpjaodan 772 . 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 772 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 103    <-> wb 104    \/ wo 682  DECID wdc 804    = wceq 1316    e. wcel 1465   A.wral 2393    u. cun 3039    C_ wss 3041   {csn 3497   suc csuc 4257   omcom 4474   "cima 4512    Fn wfn 5088   -->wf 5089   -onto->wfo 5091   ` cfv 5093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-suc 4263  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fo 5099  df-fv 5101
This theorem is referenced by:  fidcenumlemrk  6810
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