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Theorem fidcenumlemrks 6982
Description: Lemma for fidcenum 6985. Induction step for fidcenumlemrk 6983. (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 110 . . . . 5  |-  ( (
ph  /\  X  e.  ( F " J ) )  ->  X  e.  ( F " J ) )
2 elun1 3317 . . . . 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 4389 . . . . . . 7  |-  suc  J  =  ( J  u.  { J } )
54imaeq2i 4986 . . . . . 6  |-  ( F
" suc  J )  =  ( F "
( J  u.  { J } ) )
6 imaundi 5059 . . . . . 6  |-  ( F
" ( J  u.  { J } ) )  =  ( ( F
" J )  u.  ( F " { J } ) )
75, 6eqtri 2210 . . . . 5  |-  ( F
" suc  J )  =  ( ( F
" J )  u.  ( F " { J } ) )
87eleq2i 2256 . . . 4  |-  ( X  e.  ( F " suc  J )  <->  X  e.  ( ( F " J )  u.  ( F " { J }
) ) )
93, 8sylibr 134 . . 3  |-  ( (
ph  /\  X  e.  ( F " J ) )  ->  X  e.  ( F " suc  J
) )
109orcd 734 . 2  |-  ( (
ph  /\  X  e.  ( F " J ) )  ->  ( X  e.  ( F " suc  J )  \/  -.  X  e.  ( F " suc  J ) ) )
11 simpr 110 . . . . . . 7  |-  ( ( ( ph  /\  -.  X  e.  ( F " J ) )  /\  X  =  ( F `  J ) )  ->  X  =  ( F `  J ) )
12 fidcenumlemrks.x . . . . . . . . . 10  |-  ( ph  ->  X  e.  A )
13 elsng 3622 . . . . . . . . . 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 5459 . . . . . . . . . . . 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 4434 . . . . . . . . . . . . 13  |-  ( J  e.  om  ->  J  e.  suc  J )
2119, 20syl 14 . . . . . . . . . . . 12  |-  ( ph  ->  J  e.  suc  J
)
2218, 21sseldd 3171 . . . . . . . . . . 11  |-  ( ph  ->  J  e.  N )
23 fnsnfv 5596 . . . . . . . . . . 11  |-  ( ( F  Fn  N  /\  J  e.  N )  ->  { ( F `  J ) }  =  ( F " { J } ) )
2417, 22, 23syl2anc 411 . . . . . . . . . 10  |-  ( ph  ->  { ( F `  J ) }  =  ( F " { J } ) )
2524eleq2d 2259 . . . . . . . . 9  |-  ( ph  ->  ( X  e.  {
( F `  J
) }  <->  X  e.  ( F " { J } ) ) )
2614, 25bitr3d 190 . . . . . . . 8  |-  ( ph  ->  ( X  =  ( F `  J )  <-> 
X  e.  ( F
" { J }
) ) )
2726ad2antrr 488 . . . . . . 7  |-  ( ( ( ph  /\  -.  X  e.  ( F " J ) )  /\  X  =  ( F `  J ) )  -> 
( X  =  ( F `  J )  <-> 
X  e.  ( F
" { J }
) ) )
2811, 27mpbid 147 . . . . . 6  |-  ( ( ( ph  /\  -.  X  e.  ( F " J ) )  /\  X  =  ( F `  J ) )  ->  X  e.  ( F " { J } ) )
29 elun2 3318 . . . . . 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 134 . . . 4  |-  ( ( ( ph  /\  -.  X  e.  ( F " J ) )  /\  X  =  ( F `  J ) )  ->  X  e.  ( F " suc  J ) )
3231orcd 734 . . 3  |-  ( ( ( ph  /\  -.  X  e.  ( F " J ) )  /\  X  =  ( F `  J ) )  -> 
( X  e.  ( F " suc  J
)  \/  -.  X  e.  ( F " suc  J ) ) )
33 simplr 528 . . . . . . 7  |-  ( ( ( ph  /\  -.  X  e.  ( F " J ) )  /\  -.  X  =  ( F `  J )
)  ->  -.  X  e.  ( F " J
) )
34 simpr 110 . . . . . . . 8  |-  ( ( ( ph  /\  -.  X  e.  ( F " J ) )  /\  -.  X  =  ( F `  J )
)  ->  -.  X  =  ( F `  J ) )
3526ad2antrr 488 . . . . . . . 8  |-  ( ( ( ph  /\  -.  X  e.  ( F " J ) )  /\  -.  X  =  ( F `  J )
)  ->  ( X  =  ( F `  J )  <->  X  e.  ( F " { J } ) ) )
3634, 35mtbid 673 . . . . . . 7  |-  ( ( ( ph  /\  -.  X  e.  ( F " J ) )  /\  -.  X  =  ( F `  J )
)  ->  -.  X  e.  ( F " { J } ) )
37 ioran 753 . . . . . . 7  |-  ( -.  ( X  e.  ( F " J )  \/  X  e.  ( F " { J } ) )  <->  ( -.  X  e.  ( F " J )  /\  -.  X  e.  ( F " { J } ) ) )
3833, 36, 37sylanbrc 417 . . . . . 6  |-  ( ( ( ph  /\  -.  X  e.  ( F " J ) )  /\  -.  X  =  ( F `  J )
)  ->  -.  ( X  e.  ( F " J )  \/  X  e.  ( F " { J } ) ) )
39 elun 3291 . . . . . 6  |-  ( X  e.  ( ( F
" J )  u.  ( F " { J } ) )  <->  ( X  e.  ( F " J
)  \/  X  e.  ( F " { J } ) ) )
4038, 39sylnibr 678 . . . . 5  |-  ( ( ( ph  /\  -.  X  e.  ( F " J ) )  /\  -.  X  =  ( F `  J )
)  ->  -.  X  e.  ( ( F " J )  u.  ( F " { J }
) ) )
4140, 8sylnibr 678 . . . 4  |-  ( ( ( ph  /\  -.  X  e.  ( F " J ) )  /\  -.  X  =  ( F `  J )
)  ->  -.  X  e.  ( F " suc  J ) )
4241olcd 735 . . 3  |-  ( ( ( ph  /\  -.  X  e.  ( F " J ) )  /\  -.  X  =  ( F `  J )
)  ->  ( X  e.  ( F " suc  J )  \/  -.  X  e.  ( F " suc  J ) ) )
43 fof 5457 . . . . . . . 8  |-  ( F : N -onto-> A  ->  F : N --> A )
4415, 43syl 14 . . . . . . 7  |-  ( ph  ->  F : N --> A )
4544, 22ffvelcdmd 5673 . . . . . 6  |-  ( ph  ->  ( F `  J
)  e.  A )
46 fidcenumlemr.dc . . . . . 6  |-  ( ph  ->  A. x  e.  A  A. y  e.  A DECID  x  =  y )
47 eqeq1 2196 . . . . . . . 8  |-  ( x  =  X  ->  (
x  =  y  <->  X  =  y ) )
4847dcbid 839 . . . . . . 7  |-  ( x  =  X  ->  (DECID  x  =  y  <-> DECID  X  =  y )
)
49 eqeq2 2199 . . . . . . . 8  |-  ( y  =  ( F `  J )  ->  ( X  =  y  <->  X  =  ( F `  J ) ) )
5049dcbid 839 . . . . . . 7  |-  ( y  =  ( F `  J )  ->  (DECID  X  =  y  <-> DECID  X  =  ( F `  J ) ) )
5148, 50rspc2va 2870 . . . . . 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 1248 . . . . 5  |-  ( ph  -> DECID  X  =  ( F `  J ) )
53 exmiddc 837 . . . . 5  |-  (DECID  X  =  ( F `  J
)  ->  ( X  =  ( F `  J )  \/  -.  X  =  ( F `  J ) ) )
5452, 53syl 14 . . . 4  |-  ( ph  ->  ( X  =  ( F `  J )  \/  -.  X  =  ( F `  J
) ) )
5554adantr 276 . . 3  |-  ( (
ph  /\  -.  X  e.  ( F " J
) )  ->  ( X  =  ( F `  J )  \/  -.  X  =  ( F `  J ) ) )
5632, 42, 55mpjaodan 799 . 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 799 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 104    <-> wb 105    \/ wo 709  DECID wdc 835    = wceq 1364    e. wcel 2160   A.wral 2468    u. cun 3142    C_ wss 3144   {csn 3607   suc csuc 4383   omcom 4607   "cima 4647    Fn wfn 5230   -->wf 5231   -onto->wfo 5233   ` cfv 5235
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4311  df-suc 4389  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fo 5241  df-fv 5243
This theorem is referenced by:  fidcenumlemrk  6983
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