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Theorem fidcenumlemrk 6663
Description: Lemma for fidcenum 6665. (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
)
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 316 . 2  |-  ( ph  ->  ( ph  /\  K  C_  N ) )
4 sseq1 3047 . . . . 5  |-  ( w  =  (/)  ->  ( w 
C_  N  <->  (/)  C_  N
) )
54anbi2d 452 . . . 4  |-  ( w  =  (/)  ->  ( (
ph  /\  w  C_  N
)  <->  ( ph  /\  (/)  C_  N ) ) )
6 imaeq2 4770 . . . . . 6  |-  ( w  =  (/)  ->  ( F
" w )  =  ( F " (/) ) )
76eleq2d 2157 . . . . 5  |-  ( w  =  (/)  ->  ( X  e.  ( F "
w )  <->  X  e.  ( F " (/) ) ) )
87notbid 627 . . . . 5  |-  ( w  =  (/)  ->  ( -.  X  e.  ( F
" w )  <->  -.  X  e.  ( F " (/) ) ) )
97, 8orbi12d 742 . . . 4  |-  ( w  =  (/)  ->  ( ( X  e.  ( F
" w )  \/ 
-.  X  e.  ( F " w ) )  <->  ( X  e.  ( F " (/) )  \/ 
-.  X  e.  ( F " (/) ) ) ) )
105, 9imbi12d 232 . . 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 3047 . . . . 5  |-  ( w  =  j  ->  (
w  C_  N  <->  j  C_  N ) )
1211anbi2d 452 . . . 4  |-  ( w  =  j  ->  (
( ph  /\  w  C_  N )  <->  ( ph  /\  j  C_  N )
) )
13 imaeq2 4770 . . . . . 6  |-  ( w  =  j  ->  ( F " w )  =  ( F " j
) )
1413eleq2d 2157 . . . . 5  |-  ( w  =  j  ->  ( X  e.  ( F " w )  <->  X  e.  ( F " j ) ) )
1514notbid 627 . . . . 5  |-  ( w  =  j  ->  ( -.  X  e.  ( F " w )  <->  -.  X  e.  ( F " j
) ) )
1614, 15orbi12d 742 . . . 4  |-  ( w  =  j  ->  (
( X  e.  ( F " w )  \/  -.  X  e.  ( F " w
) )  <->  ( X  e.  ( F " j
)  \/  -.  X  e.  ( F " j
) ) ) )
1712, 16imbi12d 232 . . 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 3047 . . . . 5  |-  ( w  =  suc  j  -> 
( w  C_  N  <->  suc  j  C_  N )
)
1918anbi2d 452 . . . 4  |-  ( w  =  suc  j  -> 
( ( ph  /\  w  C_  N )  <->  ( ph  /\ 
suc  j  C_  N
) ) )
20 imaeq2 4770 . . . . . 6  |-  ( w  =  suc  j  -> 
( F " w
)  =  ( F
" suc  j )
)
2120eleq2d 2157 . . . . 5  |-  ( w  =  suc  j  -> 
( X  e.  ( F " w )  <-> 
X  e.  ( F
" suc  j )
) )
2221notbid 627 . . . . 5  |-  ( w  =  suc  j  -> 
( -.  X  e.  ( F " w
)  <->  -.  X  e.  ( F " suc  j
) ) )
2321, 22orbi12d 742 . . . 4  |-  ( w  =  suc  j  -> 
( ( X  e.  ( F " w
)  \/  -.  X  e.  ( F " w
) )  <->  ( X  e.  ( F " suc  j )  \/  -.  X  e.  ( F " suc  j ) ) ) )
2419, 23imbi12d 232 . . 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 3047 . . . . 5  |-  ( w  =  K  ->  (
w  C_  N  <->  K  C_  N
) )
2625anbi2d 452 . . . 4  |-  ( w  =  K  ->  (
( ph  /\  w  C_  N )  <->  ( ph  /\  K  C_  N )
) )
27 imaeq2 4770 . . . . . 6  |-  ( w  =  K  ->  ( F " w )  =  ( F " K
) )
2827eleq2d 2157 . . . . 5  |-  ( w  =  K  ->  ( X  e.  ( F " w )  <->  X  e.  ( F " K ) ) )
2928notbid 627 . . . . 5  |-  ( w  =  K  ->  ( -.  X  e.  ( F " w )  <->  -.  X  e.  ( F " K
) ) )
3028, 29orbi12d 742 . . . 4  |-  ( w  =  K  ->  (
( X  e.  ( F " w )  \/  -.  X  e.  ( F " w
) )  <->  ( X  e.  ( F " K
)  \/  -.  X  e.  ( F " K
) ) ) )
3126, 30imbi12d 232 . . 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 3290 . . . . . 6  |-  -.  X  e.  (/)
33 ima0 4791 . . . . . . 7  |-  ( F
" (/) )  =  (/)
3433eleq2i 2154 . . . . . 6  |-  ( X  e.  ( F " (/) )  <->  X  e.  (/) )
3532, 34mtbir 631 . . . . 5  |-  -.  X  e.  ( F " (/) )
3635a1i 9 . . . 4  |-  ( (
ph  /\  (/)  C_  N
)  ->  -.  X  e.  ( F " (/) ) )
3736olcd 688 . . 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 474 . . . . 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.n . . . . . 6  |-  ( ph  ->  N  e.  om )
4140ad2antrl 474 . . . . 5  |-  ( ( ( j  e.  om  /\  ( ( ph  /\  j  C_  N )  -> 
( X  e.  ( F " j )  \/  -.  X  e.  ( F " j
) ) ) )  /\  ( ph  /\  suc  j  C_  N ) )  ->  N  e.  om )
42 fidcenumlemr.f . . . . . 6  |-  ( ph  ->  F : N -onto-> A
)
4342ad2antrl 474 . . . . 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 )
44 simpll 496 . . . . 5  |-  ( ( ( j  e.  om  /\  ( ( ph  /\  j  C_  N )  -> 
( X  e.  ( F " j )  \/  -.  X  e.  ( F " j
) ) ) )  /\  ( ph  /\  suc  j  C_  N ) )  ->  j  e.  om )
45 simprr 499 . . . . 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 )
46 simprl 498 . . . . . 6  |-  ( ( ( j  e.  om  /\  ( ( ph  /\  j  C_  N )  -> 
( X  e.  ( F " j )  \/  -.  X  e.  ( F " j
) ) ) )  /\  ( ph  /\  suc  j  C_  N ) )  ->  ph )
47 sssucid 4242 . . . . . . 7  |-  j  C_  suc  j
4847, 45syl5ss 3036 . . . . . 6  |-  ( ( ( j  e.  om  /\  ( ( ph  /\  j  C_  N )  -> 
( X  e.  ( F " j )  \/  -.  X  e.  ( F " j
) ) ) )  /\  ( ph  /\  suc  j  C_  N ) )  ->  j  C_  N )
49 simplr 497 . . . . . 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
) ) ) )
5046, 48, 49mp2and 424 . . . . 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
) ) )
51 fidcenumlemrk.x . . . . . 6  |-  ( ph  ->  X  e.  A )
5251ad2antrl 474 . . . . 5  |-  ( ( ( j  e.  om  /\  ( ( ph  /\  j  C_  N )  -> 
( X  e.  ( F " j )  \/  -.  X  e.  ( F " j
) ) ) )  /\  ( ph  /\  suc  j  C_  N ) )  ->  X  e.  A )
5339, 41, 43, 44, 45, 50, 52fidcenumlemrks 6662 . . . 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 ) ) )
5453exp31 356 . . 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 ) ) ) ) )
5510, 17, 24, 31, 37, 54finds 4415 . 2  |-  ( K  e.  om  ->  (
( ph  /\  K  C_  N )  ->  ( X  e.  ( F " K )  \/  -.  X  e.  ( F " K ) ) ) )
561, 3, 55sylc 61 1  |-  ( ph  ->  ( X  e.  ( F " K )  \/  -.  X  e.  ( F " K
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    \/ wo 664  DECID wdc 780    = wceq 1289    e. wcel 1438   A.wral 2359    C_ wss 2999   (/)c0 3286   suc csuc 4192   omcom 4405   "cima 4441   -onto->wfo 5013
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-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-iinf 4403
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3an 926  df-tru 1292  df-fal 1295  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-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-br 3846  df-opab 3900  df-id 4120  df-suc 4198  df-iom 4406  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:  fidcenumlemr  6664
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