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Theorem fin1a2lem6 8031
Description: Lemma for fin1a2 8041. Establish that  om can be broken into two equipollent pieces. (Contributed by Stefan O'Rear, 7-Nov-2014.)
Hypotheses
Ref Expression
fin1a2lem.b  |-  E  =  ( x  e.  om  |->  ( 2o  .o  x
) )
fin1a2lem.aa  |-  S  =  ( x  e.  On  |->  suc  x )
Assertion
Ref Expression
fin1a2lem6  |-  ( S  |`  ran  E ) : ran  E -1-1-onto-> ( om  \  ran  E )

Proof of Theorem fin1a2lem6
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fin1a2lem.aa . . . 4  |-  S  =  ( x  e.  On  |->  suc  x )
21fin1a2lem2 8027 . . 3  |-  S : On
-1-1-> On
3 fin1a2lem.b . . . . 5  |-  E  =  ( x  e.  om  |->  ( 2o  .o  x
) )
43fin1a2lem4 8029 . . . 4  |-  E : om
-1-1-> om
5 f1f 5437 . . . 4  |-  ( E : om -1-1-> om  ->  E : om --> om )
6 frn 5395 . . . . 5  |-  ( E : om --> om  ->  ran 
E  C_  om )
7 omsson 4660 . . . . 5  |-  om  C_  On
86, 7syl6ss 3191 . . . 4  |-  ( E : om --> om  ->  ran 
E  C_  On )
94, 5, 8mp2b 9 . . 3  |-  ran  E  C_  On
10 f1ores 5487 . . 3  |-  ( ( S : On -1-1-> On  /\ 
ran  E  C_  On )  ->  ( S  |`  ran  E ) : ran  E -1-1-onto-> ( S " ran  E
) )
112, 9, 10mp2an 653 . 2  |-  ( S  |`  ran  E ) : ran  E -1-1-onto-> ( S " ran  E )
129sseli 3176 . . . . . . . . 9  |-  ( b  e.  ran  E  -> 
b  e.  On )
131fin1a2lem1 8026 . . . . . . . . 9  |-  ( b  e.  On  ->  ( S `  b )  =  suc  b )
1412, 13syl 15 . . . . . . . 8  |-  ( b  e.  ran  E  -> 
( S `  b
)  =  suc  b
)
1514eqeq1d 2291 . . . . . . 7  |-  ( b  e.  ran  E  -> 
( ( S `  b )  =  a  <->  suc  b  =  a
) )
1615rexbiia 2576 . . . . . 6  |-  ( E. b  e.  ran  E
( S `  b
)  =  a  <->  E. b  e.  ran  E  suc  b  =  a )
174, 5, 6mp2b 9 . . . . . . . . . . . 12  |-  ran  E  C_ 
om
1817sseli 3176 . . . . . . . . . . 11  |-  ( b  e.  ran  E  -> 
b  e.  om )
19 peano2 4676 . . . . . . . . . . 11  |-  ( b  e.  om  ->  suc  b  e.  om )
2018, 19syl 15 . . . . . . . . . 10  |-  ( b  e.  ran  E  ->  suc  b  e.  om )
213fin1a2lem5 8030 . . . . . . . . . . . 12  |-  ( b  e.  om  ->  (
b  e.  ran  E  <->  -. 
suc  b  e.  ran  E ) )
2221biimpd 198 . . . . . . . . . . 11  |-  ( b  e.  om  ->  (
b  e.  ran  E  ->  -.  suc  b  e. 
ran  E ) )
2318, 22mpcom 32 . . . . . . . . . 10  |-  ( b  e.  ran  E  ->  -.  suc  b  e.  ran  E )
2420, 23jca 518 . . . . . . . . 9  |-  ( b  e.  ran  E  -> 
( suc  b  e.  om 
/\  -.  suc  b  e. 
ran  E ) )
25 eleq1 2343 . . . . . . . . . 10  |-  ( suc  b  =  a  -> 
( suc  b  e.  om  <->  a  e.  om ) )
26 eleq1 2343 . . . . . . . . . . 11  |-  ( suc  b  =  a  -> 
( suc  b  e.  ran  E  <->  a  e.  ran  E ) )
2726notbid 285 . . . . . . . . . 10  |-  ( suc  b  =  a  -> 
( -.  suc  b  e.  ran  E  <->  -.  a  e.  ran  E ) )
2825, 27anbi12d 691 . . . . . . . . 9  |-  ( suc  b  =  a  -> 
( ( suc  b  e.  om  /\  -.  suc  b  e.  ran  E )  <-> 
( a  e.  om  /\ 
-.  a  e.  ran  E ) ) )
2924, 28syl5ibcom 211 . . . . . . . 8  |-  ( b  e.  ran  E  -> 
( suc  b  =  a  ->  ( a  e. 
om  /\  -.  a  e.  ran  E ) ) )
3029rexlimiv 2661 . . . . . . 7  |-  ( E. b  e.  ran  E  suc  b  =  a  ->  ( a  e.  om  /\ 
-.  a  e.  ran  E ) )
31 peano1 4675 . . . . . . . . . . . . . 14  |-  (/)  e.  om
323fin1a2lem3 8028 . . . . . . . . . . . . . 14  |-  ( (/)  e.  om  ->  ( E `  (/) )  =  ( 2o  .o  (/) ) )
3331, 32ax-mp 8 . . . . . . . . . . . . 13  |-  ( E `
 (/) )  =  ( 2o  .o  (/) )
34 om0x 6518 . . . . . . . . . . . . 13  |-  ( 2o 
.o  (/) )  =  (/)
3533, 34eqtri 2303 . . . . . . . . . . . 12  |-  ( E `
 (/) )  =  (/)
36 f1fun 5439 . . . . . . . . . . . . . 14  |-  ( E : om -1-1-> om  ->  Fun 
E )
374, 36ax-mp 8 . . . . . . . . . . . . 13  |-  Fun  E
38 f1dm 5441 . . . . . . . . . . . . . . 15  |-  ( E : om -1-1-> om  ->  dom 
E  =  om )
394, 38ax-mp 8 . . . . . . . . . . . . . 14  |-  dom  E  =  om
4031, 39eleqtrri 2356 . . . . . . . . . . . . 13  |-  (/)  e.  dom  E
41 fvelrn 5661 . . . . . . . . . . . . 13  |-  ( ( Fun  E  /\  (/)  e.  dom  E )  ->  ( E `  (/) )  e.  ran  E )
4237, 40, 41mp2an 653 . . . . . . . . . . . 12  |-  ( E `
 (/) )  e.  ran  E
4335, 42eqeltrri 2354 . . . . . . . . . . 11  |-  (/)  e.  ran  E
44 eleq1 2343 . . . . . . . . . . 11  |-  ( a  =  (/)  ->  ( a  e.  ran  E  <->  (/)  e.  ran  E ) )
4543, 44mpbiri 224 . . . . . . . . . 10  |-  ( a  =  (/)  ->  a  e. 
ran  E )
4645necon3bi 2487 . . . . . . . . 9  |-  ( -.  a  e.  ran  E  ->  a  =/=  (/) )
47 nnsuc 4673 . . . . . . . . 9  |-  ( ( a  e.  om  /\  a  =/=  (/) )  ->  E. b  e.  om  a  =  suc  b )
4846, 47sylan2 460 . . . . . . . 8  |-  ( ( a  e.  om  /\  -.  a  e.  ran  E )  ->  E. b  e.  om  a  =  suc  b )
49 eleq1 2343 . . . . . . . . . . . . . . . 16  |-  ( a  =  suc  b  -> 
( a  e.  om  <->  suc  b  e.  om )
)
50 eleq1 2343 . . . . . . . . . . . . . . . . 17  |-  ( a  =  suc  b  -> 
( a  e.  ran  E  <->  suc  b  e.  ran  E ) )
5150notbid 285 . . . . . . . . . . . . . . . 16  |-  ( a  =  suc  b  -> 
( -.  a  e. 
ran  E  <->  -.  suc  b  e. 
ran  E ) )
5249, 51anbi12d 691 . . . . . . . . . . . . . . 15  |-  ( a  =  suc  b  -> 
( ( a  e. 
om  /\  -.  a  e.  ran  E )  <->  ( suc  b  e.  om  /\  -.  suc  b  e.  ran  E ) ) )
5352anbi1d 685 . . . . . . . . . . . . . 14  |-  ( a  =  suc  b  -> 
( ( ( a  e.  om  /\  -.  a  e.  ran  E )  /\  b  e.  om ) 
<->  ( ( suc  b  e.  om  /\  -.  suc  b  e.  ran  E )  /\  b  e.  om ) ) )
54 simplr 731 . . . . . . . . . . . . . . 15  |-  ( ( ( suc  b  e. 
om  /\  -.  suc  b  e.  ran  E )  /\  b  e.  om )  ->  -.  suc  b  e. 
ran  E )
5521adantl 452 . . . . . . . . . . . . . . 15  |-  ( ( ( suc  b  e. 
om  /\  -.  suc  b  e.  ran  E )  /\  b  e.  om )  ->  ( b  e.  ran  E  <->  -.  suc  b  e.  ran  E ) )
5654, 55mpbird 223 . . . . . . . . . . . . . 14  |-  ( ( ( suc  b  e. 
om  /\  -.  suc  b  e.  ran  E )  /\  b  e.  om )  ->  b  e.  ran  E
)
5753, 56syl6bi 219 . . . . . . . . . . . . 13  |-  ( a  =  suc  b  -> 
( ( ( a  e.  om  /\  -.  a  e.  ran  E )  /\  b  e.  om )  ->  b  e.  ran  E ) )
5857com12 27 . . . . . . . . . . . 12  |-  ( ( ( a  e.  om  /\ 
-.  a  e.  ran  E )  /\  b  e. 
om )  ->  (
a  =  suc  b  ->  b  e.  ran  E
) )
5958impr 602 . . . . . . . . . . 11  |-  ( ( ( a  e.  om  /\ 
-.  a  e.  ran  E )  /\  ( b  e.  om  /\  a  =  suc  b ) )  ->  b  e.  ran  E )
60 simprr 733 . . . . . . . . . . . 12  |-  ( ( ( a  e.  om  /\ 
-.  a  e.  ran  E )  /\  ( b  e.  om  /\  a  =  suc  b ) )  ->  a  =  suc  b )
6160eqcomd 2288 . . . . . . . . . . 11  |-  ( ( ( a  e.  om  /\ 
-.  a  e.  ran  E )  /\  ( b  e.  om  /\  a  =  suc  b ) )  ->  suc  b  =  a )
6259, 61jca 518 . . . . . . . . . 10  |-  ( ( ( a  e.  om  /\ 
-.  a  e.  ran  E )  /\  ( b  e.  om  /\  a  =  suc  b ) )  ->  ( b  e. 
ran  E  /\  suc  b  =  a ) )
6362ex 423 . . . . . . . . 9  |-  ( ( a  e.  om  /\  -.  a  e.  ran  E )  ->  ( (
b  e.  om  /\  a  =  suc  b )  ->  ( b  e. 
ran  E  /\  suc  b  =  a ) ) )
6463reximdv2 2652 . . . . . . . 8  |-  ( ( a  e.  om  /\  -.  a  e.  ran  E )  ->  ( E. b  e.  om  a  =  suc  b  ->  E. b  e.  ran  E  suc  b  =  a ) )
6548, 64mpd 14 . . . . . . 7  |-  ( ( a  e.  om  /\  -.  a  e.  ran  E )  ->  E. b  e.  ran  E  suc  b  =  a )
6630, 65impbii 180 . . . . . 6  |-  ( E. b  e.  ran  E  suc  b  =  a  <->  ( a  e.  om  /\  -.  a  e.  ran  E ) )
6716, 66bitri 240 . . . . 5  |-  ( E. b  e.  ran  E
( S `  b
)  =  a  <->  ( a  e.  om  /\  -.  a  e.  ran  E ) )
68 f1fn 5438 . . . . . . 7  |-  ( S : On -1-1-> On  ->  S  Fn  On )
692, 68ax-mp 8 . . . . . 6  |-  S  Fn  On
70 fvelimab 5578 . . . . . 6  |-  ( ( S  Fn  On  /\  ran  E  C_  On )  ->  ( a  e.  ( S " ran  E
)  <->  E. b  e.  ran  E ( S `  b
)  =  a ) )
7169, 9, 70mp2an 653 . . . . 5  |-  ( a  e.  ( S " ran  E )  <->  E. b  e.  ran  E ( S `
 b )  =  a )
72 eldif 3162 . . . . 5  |-  ( a  e.  ( om  \  ran  E )  <->  ( a  e. 
om  /\  -.  a  e.  ran  E ) )
7367, 71, 723bitr4i 268 . . . 4  |-  ( a  e.  ( S " ran  E )  <->  a  e.  ( om  \  ran  E
) )
7473eqriv 2280 . . 3  |-  ( S
" ran  E )  =  ( om  \  ran  E )
75 f1oeq3 5465 . . 3  |-  ( ( S " ran  E
)  =  ( om 
\  ran  E )  ->  ( ( S  |`  ran  E ) : ran  E -1-1-onto-> ( S " ran  E
)  <->  ( S  |`  ran  E ) : ran  E -1-1-onto-> ( om  \  ran  E
) ) )
7674, 75ax-mp 8 . 2  |-  ( ( S  |`  ran  E ) : ran  E -1-1-onto-> ( S
" ran  E )  <->  ( S  |`  ran  E ) : ran  E -1-1-onto-> ( om 
\  ran  E )
)
7711, 76mpbi 199 1  |-  ( S  |`  ran  E ) : ran  E -1-1-onto-> ( om  \  ran  E )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544    \ cdif 3149    C_ wss 3152   (/)c0 3455    e. cmpt 4077   Oncon0 4392   suc csuc 4394   omcom 4656   dom cdm 4689   ran crn 4690    |` cres 4691   "cima 4692   Fun wfun 5249    Fn wfn 5250   -->wf 5251   -1-1->wf1 5252   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   2oc2o 6473    .o comu 6477
This theorem is referenced by:  fin1a2lem7  8032
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-omul 6484
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