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Theorem fidcenumlemrk 6846
Description: Lemma for fidcenum 6848. (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
)
fidcenumlemrk.k  |-  ( ph  ->  K  e.  om )
fidcenumlemrk.kn  |-  ( ph  ->  K  C_  N )
fidcenumlemrk.x  |-  ( ph  ->  X  e.  A )
Assertion
Ref Expression
fidcenumlemrk  |-  ( ph  ->  ( X  e.  ( F " K )  \/  -.  X  e.  ( F " K
) ) )
Distinct variable groups:    x, F, y   
x, X, y    x, A, y
Allowed substitution hints:    ph( x, y)    K( x, y)    N( x, y)

Proof of Theorem fidcenumlemrk
Dummy variables  j  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fidcenumlemrk.k . 2  |-  ( ph  ->  K  e.  om )
2 fidcenumlemrk.kn . . 3  |-  ( ph  ->  K  C_  N )
32ancli 321 . 2  |-  ( ph  ->  ( ph  /\  K  C_  N ) )
4 sseq1 3121 . . . . 5  |-  ( w  =  (/)  ->  ( w 
C_  N  <->  (/)  C_  N
) )
54anbi2d 460 . . . 4  |-  ( w  =  (/)  ->  ( (
ph  /\  w  C_  N
)  <->  ( ph  /\  (/)  C_  N ) ) )
6 imaeq2 4881 . . . . . 6  |-  ( w  =  (/)  ->  ( F
" w )  =  ( F " (/) ) )
76eleq2d 2210 . . . . 5  |-  ( w  =  (/)  ->  ( X  e.  ( F "
w )  <->  X  e.  ( F " (/) ) ) )
87notbid 657 . . . . 5  |-  ( w  =  (/)  ->  ( -.  X  e.  ( F
" w )  <->  -.  X  e.  ( F " (/) ) ) )
97, 8orbi12d 783 . . . 4  |-  ( w  =  (/)  ->  ( ( X  e.  ( F
" w )  \/ 
-.  X  e.  ( F " w ) )  <->  ( X  e.  ( F " (/) )  \/ 
-.  X  e.  ( F " (/) ) ) ) )
105, 9imbi12d 233 . . 3  |-  ( w  =  (/)  ->  ( ( ( ph  /\  w  C_  N )  ->  ( X  e.  ( F " w )  \/  -.  X  e.  ( F " w ) ) )  <-> 
( ( ph  /\  (/)  C_  N )  ->  ( X  e.  ( F "
(/) )  \/  -.  X  e.  ( F "
(/) ) ) ) ) )
11 sseq1 3121 . . . . 5  |-  ( w  =  j  ->  (
w  C_  N  <->  j  C_  N ) )
1211anbi2d 460 . . . 4  |-  ( w  =  j  ->  (
( ph  /\  w  C_  N )  <->  ( ph  /\  j  C_  N )
) )
13 imaeq2 4881 . . . . . 6  |-  ( w  =  j  ->  ( F " w )  =  ( F " j
) )
1413eleq2d 2210 . . . . 5  |-  ( w  =  j  ->  ( X  e.  ( F " w )  <->  X  e.  ( F " j ) ) )
1514notbid 657 . . . . 5  |-  ( w  =  j  ->  ( -.  X  e.  ( F " w )  <->  -.  X  e.  ( F " j
) ) )
1614, 15orbi12d 783 . . . 4  |-  ( w  =  j  ->  (
( X  e.  ( F " w )  \/  -.  X  e.  ( F " w
) )  <->  ( X  e.  ( F " j
)  \/  -.  X  e.  ( F " j
) ) ) )
1712, 16imbi12d 233 . . 3  |-  ( w  =  j  ->  (
( ( ph  /\  w  C_  N )  -> 
( X  e.  ( F " w )  \/  -.  X  e.  ( F " w
) ) )  <->  ( ( ph  /\  j  C_  N
)  ->  ( X  e.  ( F " j
)  \/  -.  X  e.  ( F " j
) ) ) ) )
18 sseq1 3121 . . . . 5  |-  ( w  =  suc  j  -> 
( w  C_  N  <->  suc  j  C_  N )
)
1918anbi2d 460 . . . 4  |-  ( w  =  suc  j  -> 
( ( ph  /\  w  C_  N )  <->  ( ph  /\ 
suc  j  C_  N
) ) )
20 imaeq2 4881 . . . . . 6  |-  ( w  =  suc  j  -> 
( F " w
)  =  ( F
" suc  j )
)
2120eleq2d 2210 . . . . 5  |-  ( w  =  suc  j  -> 
( X  e.  ( F " w )  <-> 
X  e.  ( F
" suc  j )
) )
2221notbid 657 . . . . 5  |-  ( w  =  suc  j  -> 
( -.  X  e.  ( F " w
)  <->  -.  X  e.  ( F " suc  j
) ) )
2321, 22orbi12d 783 . . . 4  |-  ( w  =  suc  j  -> 
( ( X  e.  ( F " w
)  \/  -.  X  e.  ( F " w
) )  <->  ( X  e.  ( F " suc  j )  \/  -.  X  e.  ( F " suc  j ) ) ) )
2419, 23imbi12d 233 . . 3  |-  ( w  =  suc  j  -> 
( ( ( ph  /\  w  C_  N )  ->  ( X  e.  ( F " w )  \/  -.  X  e.  ( F " w
) ) )  <->  ( ( ph  /\  suc  j  C_  N )  ->  ( X  e.  ( F " suc  j )  \/ 
-.  X  e.  ( F " suc  j
) ) ) ) )
25 sseq1 3121 . . . . 5  |-  ( w  =  K  ->  (
w  C_  N  <->  K  C_  N
) )
2625anbi2d 460 . . . 4  |-  ( w  =  K  ->  (
( ph  /\  w  C_  N )  <->  ( ph  /\  K  C_  N )
) )
27 imaeq2 4881 . . . . . 6  |-  ( w  =  K  ->  ( F " w )  =  ( F " K
) )
2827eleq2d 2210 . . . . 5  |-  ( w  =  K  ->  ( X  e.  ( F " w )  <->  X  e.  ( F " K ) ) )
2928notbid 657 . . . . 5  |-  ( w  =  K  ->  ( -.  X  e.  ( F " w )  <->  -.  X  e.  ( F " K
) ) )
3028, 29orbi12d 783 . . . 4  |-  ( w  =  K  ->  (
( X  e.  ( F " w )  \/  -.  X  e.  ( F " w
) )  <->  ( X  e.  ( F " K
)  \/  -.  X  e.  ( F " K
) ) ) )
3126, 30imbi12d 233 . . 3  |-  ( w  =  K  ->  (
( ( ph  /\  w  C_  N )  -> 
( X  e.  ( F " w )  \/  -.  X  e.  ( F " w
) ) )  <->  ( ( ph  /\  K  C_  N
)  ->  ( X  e.  ( F " K
)  \/  -.  X  e.  ( F " K
) ) ) ) )
32 noel 3368 . . . . . 6  |-  -.  X  e.  (/)
33 ima0 4902 . . . . . . 7  |-  ( F
" (/) )  =  (/)
3433eleq2i 2207 . . . . . 6  |-  ( X  e.  ( F " (/) )  <->  X  e.  (/) )
3532, 34mtbir 661 . . . . 5  |-  -.  X  e.  ( F " (/) )
3635a1i 9 . . . 4  |-  ( (
ph  /\  (/)  C_  N
)  ->  -.  X  e.  ( F " (/) ) )
3736olcd 724 . . 3  |-  ( (
ph  /\  (/)  C_  N
)  ->  ( X  e.  ( F " (/) )  \/ 
-.  X  e.  ( F " (/) ) ) )
38 fidcenumlemr.dc . . . . . 6  |-  ( ph  ->  A. x  e.  A  A. y  e.  A DECID  x  =  y )
3938ad2antrl 482 . . . . 5  |-  ( ( ( j  e.  om  /\  ( ( ph  /\  j  C_  N )  -> 
( X  e.  ( F " j )  \/  -.  X  e.  ( F " j
) ) ) )  /\  ( ph  /\  suc  j  C_  N ) )  ->  A. x  e.  A  A. y  e.  A DECID  x  =  y
)
40 fidcenumlemr.f . . . . . 6  |-  ( ph  ->  F : N -onto-> A
)
4140ad2antrl 482 . . . . 5  |-  ( ( ( j  e.  om  /\  ( ( ph  /\  j  C_  N )  -> 
( X  e.  ( F " j )  \/  -.  X  e.  ( F " j
) ) ) )  /\  ( ph  /\  suc  j  C_  N ) )  ->  F : N -onto-> A )
42 simpll 519 . . . . 5  |-  ( ( ( j  e.  om  /\  ( ( ph  /\  j  C_  N )  -> 
( X  e.  ( F " j )  \/  -.  X  e.  ( F " j
) ) ) )  /\  ( ph  /\  suc  j  C_  N ) )  ->  j  e.  om )
43 simprr 522 . . . . 5  |-  ( ( ( j  e.  om  /\  ( ( ph  /\  j  C_  N )  -> 
( X  e.  ( F " j )  \/  -.  X  e.  ( F " j
) ) ) )  /\  ( ph  /\  suc  j  C_  N ) )  ->  suc  j  C_  N )
44 simprl 521 . . . . . 6  |-  ( ( ( j  e.  om  /\  ( ( ph  /\  j  C_  N )  -> 
( X  e.  ( F " j )  \/  -.  X  e.  ( F " j
) ) ) )  /\  ( ph  /\  suc  j  C_  N ) )  ->  ph )
45 sssucid 4341 . . . . . . 7  |-  j  C_  suc  j
4645, 43sstrid 3109 . . . . . 6  |-  ( ( ( j  e.  om  /\  ( ( ph  /\  j  C_  N )  -> 
( X  e.  ( F " j )  \/  -.  X  e.  ( F " j
) ) ) )  /\  ( ph  /\  suc  j  C_  N ) )  ->  j  C_  N )
47 simplr 520 . . . . . 6  |-  ( ( ( j  e.  om  /\  ( ( ph  /\  j  C_  N )  -> 
( X  e.  ( F " j )  \/  -.  X  e.  ( F " j
) ) ) )  /\  ( ph  /\  suc  j  C_  N ) )  ->  ( ( ph  /\  j  C_  N
)  ->  ( X  e.  ( F " j
)  \/  -.  X  e.  ( F " j
) ) ) )
4844, 46, 47mp2and 430 . . . . 5  |-  ( ( ( j  e.  om  /\  ( ( ph  /\  j  C_  N )  -> 
( X  e.  ( F " j )  \/  -.  X  e.  ( F " j
) ) ) )  /\  ( ph  /\  suc  j  C_  N ) )  ->  ( X  e.  ( F " j
)  \/  -.  X  e.  ( F " j
) ) )
49 fidcenumlemrk.x . . . . . 6  |-  ( ph  ->  X  e.  A )
5049ad2antrl 482 . . . . 5  |-  ( ( ( j  e.  om  /\  ( ( ph  /\  j  C_  N )  -> 
( X  e.  ( F " j )  \/  -.  X  e.  ( F " j
) ) ) )  /\  ( ph  /\  suc  j  C_  N ) )  ->  X  e.  A )
5139, 41, 42, 43, 48, 50fidcenumlemrks 6845 . . . 4  |-  ( ( ( j  e.  om  /\  ( ( ph  /\  j  C_  N )  -> 
( X  e.  ( F " j )  \/  -.  X  e.  ( F " j
) ) ) )  /\  ( ph  /\  suc  j  C_  N ) )  ->  ( X  e.  ( F " suc  j )  \/  -.  X  e.  ( F " suc  j ) ) )
5251exp31 362 . . 3  |-  ( j  e.  om  ->  (
( ( ph  /\  j  C_  N )  -> 
( X  e.  ( F " j )  \/  -.  X  e.  ( F " j
) ) )  -> 
( ( ph  /\  suc  j  C_  N )  ->  ( X  e.  ( F " suc  j )  \/  -.  X  e.  ( F " suc  j ) ) ) ) )
5310, 17, 24, 31, 37, 52finds 4518 . 2  |-  ( K  e.  om  ->  (
( ph  /\  K  C_  N )  ->  ( X  e.  ( F " K )  \/  -.  X  e.  ( F " K ) ) ) )
541, 3, 53sylc 62 1  |-  ( ph  ->  ( X  e.  ( F " K )  \/  -.  X  e.  ( F " K
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    \/ wo 698  DECID wdc 820    = wceq 1332    e. wcel 1481   A.wral 2417    C_ wss 3072   (/)c0 3364   suc csuc 4291   omcom 4508   "cima 4546   -onto->wfo 5125
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4050  ax-nul 4058  ax-pow 4102  ax-pr 4135  ax-un 4359  ax-iinf 4506
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2689  df-sbc 2911  df-dif 3074  df-un 3076  df-in 3078  df-ss 3085  df-nul 3365  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-uni 3741  df-int 3776  df-br 3934  df-opab 3994  df-id 4219  df-suc 4297  df-iom 4509  df-xp 4549  df-rel 4550  df-cnv 4551  df-co 4552  df-dm 4553  df-rn 4554  df-res 4555  df-ima 4556  df-iota 5092  df-fun 5129  df-fn 5130  df-f 5131  df-fo 5133  df-fv 5135
This theorem is referenced by:  fidcenumlemr  6847  ennnfonelemdc  11939
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