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Theorem fin1a2lem6 8287
Description: Lemma for fin1a2 8297. 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 8283 . . 3  |-  S : On
-1-1-> On
3 fin1a2lem.b . . . . 5  |-  E  =  ( x  e.  om  |->  ( 2o  .o  x
) )
43fin1a2lem4 8285 . . . 4  |-  E : om
-1-1-> om
5 f1f 5641 . . . 4  |-  ( E : om -1-1-> om  ->  E : om --> om )
6 frn 5599 . . . . 5  |-  ( E : om --> om  ->  ran 
E  C_  om )
7 omsson 4851 . . . . 5  |-  om  C_  On
86, 7syl6ss 3362 . . . 4  |-  ( E : om --> om  ->  ran 
E  C_  On )
94, 5, 8mp2b 10 . . 3  |-  ran  E  C_  On
10 f1ores 5691 . . 3  |-  ( ( S : On -1-1-> On  /\ 
ran  E  C_  On )  ->  ( S  |`  ran  E ) : ran  E -1-1-onto-> ( S " ran  E
) )
112, 9, 10mp2an 655 . 2  |-  ( S  |`  ran  E ) : ran  E -1-1-onto-> ( S " ran  E )
129sseli 3346 . . . . . . . . 9  |-  ( b  e.  ran  E  -> 
b  e.  On )
131fin1a2lem1 8282 . . . . . . . . 9  |-  ( b  e.  On  ->  ( S `  b )  =  suc  b )
1412, 13syl 16 . . . . . . . 8  |-  ( b  e.  ran  E  -> 
( S `  b
)  =  suc  b
)
1514eqeq1d 2446 . . . . . . 7  |-  ( b  e.  ran  E  -> 
( ( S `  b )  =  a  <->  suc  b  =  a
) )
1615rexbiia 2740 . . . . . 6  |-  ( E. b  e.  ran  E
( S `  b
)  =  a  <->  E. b  e.  ran  E  suc  b  =  a )
174, 5, 6mp2b 10 . . . . . . . . . . . 12  |-  ran  E  C_ 
om
1817sseli 3346 . . . . . . . . . . 11  |-  ( b  e.  ran  E  -> 
b  e.  om )
19 peano2 4867 . . . . . . . . . . 11  |-  ( b  e.  om  ->  suc  b  e.  om )
2018, 19syl 16 . . . . . . . . . 10  |-  ( b  e.  ran  E  ->  suc  b  e.  om )
213fin1a2lem5 8286 . . . . . . . . . . . 12  |-  ( b  e.  om  ->  (
b  e.  ran  E  <->  -. 
suc  b  e.  ran  E ) )
2221biimpd 200 . . . . . . . . . . 11  |-  ( b  e.  om  ->  (
b  e.  ran  E  ->  -.  suc  b  e. 
ran  E ) )
2318, 22mpcom 35 . . . . . . . . . 10  |-  ( b  e.  ran  E  ->  -.  suc  b  e.  ran  E )
2420, 23jca 520 . . . . . . . . 9  |-  ( b  e.  ran  E  -> 
( suc  b  e.  om 
/\  -.  suc  b  e. 
ran  E ) )
25 eleq1 2498 . . . . . . . . . 10  |-  ( suc  b  =  a  -> 
( suc  b  e.  om  <->  a  e.  om ) )
26 eleq1 2498 . . . . . . . . . . 11  |-  ( suc  b  =  a  -> 
( suc  b  e.  ran  E  <->  a  e.  ran  E ) )
2726notbid 287 . . . . . . . . . 10  |-  ( suc  b  =  a  -> 
( -.  suc  b  e.  ran  E  <->  -.  a  e.  ran  E ) )
2825, 27anbi12d 693 . . . . . . . . 9  |-  ( suc  b  =  a  -> 
( ( suc  b  e.  om  /\  -.  suc  b  e.  ran  E )  <-> 
( a  e.  om  /\ 
-.  a  e.  ran  E ) ) )
2924, 28syl5ibcom 213 . . . . . . . 8  |-  ( b  e.  ran  E  -> 
( suc  b  =  a  ->  ( a  e. 
om  /\  -.  a  e.  ran  E ) ) )
3029rexlimiv 2826 . . . . . . 7  |-  ( E. b  e.  ran  E  suc  b  =  a  ->  ( a  e.  om  /\ 
-.  a  e.  ran  E ) )
31 peano1 4866 . . . . . . . . . . . . . 14  |-  (/)  e.  om
323fin1a2lem3 8284 . . . . . . . . . . . . . 14  |-  ( (/)  e.  om  ->  ( E `  (/) )  =  ( 2o  .o  (/) ) )
3331, 32ax-mp 8 . . . . . . . . . . . . 13  |-  ( E `
 (/) )  =  ( 2o  .o  (/) )
34 om0x 6765 . . . . . . . . . . . . 13  |-  ( 2o 
.o  (/) )  =  (/)
3533, 34eqtri 2458 . . . . . . . . . . . 12  |-  ( E `
 (/) )  =  (/)
36 f1fun 5643 . . . . . . . . . . . . . 14  |-  ( E : om -1-1-> om  ->  Fun 
E )
374, 36ax-mp 8 . . . . . . . . . . . . 13  |-  Fun  E
38 f1dm 5645 . . . . . . . . . . . . . . 15  |-  ( E : om -1-1-> om  ->  dom 
E  =  om )
394, 38ax-mp 8 . . . . . . . . . . . . . 14  |-  dom  E  =  om
4031, 39eleqtrri 2511 . . . . . . . . . . . . 13  |-  (/)  e.  dom  E
41 fvelrn 5868 . . . . . . . . . . . . 13  |-  ( ( Fun  E  /\  (/)  e.  dom  E )  ->  ( E `  (/) )  e.  ran  E )
4237, 40, 41mp2an 655 . . . . . . . . . . . 12  |-  ( E `
 (/) )  e.  ran  E
4335, 42eqeltrri 2509 . . . . . . . . . . 11  |-  (/)  e.  ran  E
44 eleq1 2498 . . . . . . . . . . 11  |-  ( a  =  (/)  ->  ( a  e.  ran  E  <->  (/)  e.  ran  E ) )
4543, 44mpbiri 226 . . . . . . . . . 10  |-  ( a  =  (/)  ->  a  e. 
ran  E )
4645necon3bi 2647 . . . . . . . . 9  |-  ( -.  a  e.  ran  E  ->  a  =/=  (/) )
47 nnsuc 4864 . . . . . . . . 9  |-  ( ( a  e.  om  /\  a  =/=  (/) )  ->  E. b  e.  om  a  =  suc  b )
4846, 47sylan2 462 . . . . . . . 8  |-  ( ( a  e.  om  /\  -.  a  e.  ran  E )  ->  E. b  e.  om  a  =  suc  b )
49 eleq1 2498 . . . . . . . . . . . . . . . 16  |-  ( a  =  suc  b  -> 
( a  e.  om  <->  suc  b  e.  om )
)
50 eleq1 2498 . . . . . . . . . . . . . . . . 17  |-  ( a  =  suc  b  -> 
( a  e.  ran  E  <->  suc  b  e.  ran  E ) )
5150notbid 287 . . . . . . . . . . . . . . . 16  |-  ( a  =  suc  b  -> 
( -.  a  e. 
ran  E  <->  -.  suc  b  e. 
ran  E ) )
5249, 51anbi12d 693 . . . . . . . . . . . . . . 15  |-  ( a  =  suc  b  -> 
( ( a  e. 
om  /\  -.  a  e.  ran  E )  <->  ( suc  b  e.  om  /\  -.  suc  b  e.  ran  E ) ) )
5352anbi1d 687 . . . . . . . . . . . . . 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 733 . . . . . . . . . . . . . . 15  |-  ( ( ( suc  b  e. 
om  /\  -.  suc  b  e.  ran  E )  /\  b  e.  om )  ->  -.  suc  b  e. 
ran  E )
5521adantl 454 . . . . . . . . . . . . . . 15  |-  ( ( ( suc  b  e. 
om  /\  -.  suc  b  e.  ran  E )  /\  b  e.  om )  ->  ( b  e.  ran  E  <->  -.  suc  b  e.  ran  E ) )
5654, 55mpbird 225 . . . . . . . . . . . . . 14  |-  ( ( ( suc  b  e. 
om  /\  -.  suc  b  e.  ran  E )  /\  b  e.  om )  ->  b  e.  ran  E
)
5753, 56syl6bi 221 . . . . . . . . . . . . 13  |-  ( a  =  suc  b  -> 
( ( ( a  e.  om  /\  -.  a  e.  ran  E )  /\  b  e.  om )  ->  b  e.  ran  E ) )
5857com12 30 . . . . . . . . . . . 12  |-  ( ( ( a  e.  om  /\ 
-.  a  e.  ran  E )  /\  b  e. 
om )  ->  (
a  =  suc  b  ->  b  e.  ran  E
) )
5958impr 604 . . . . . . . . . . 11  |-  ( ( ( a  e.  om  /\ 
-.  a  e.  ran  E )  /\  ( b  e.  om  /\  a  =  suc  b ) )  ->  b  e.  ran  E )
60 simprr 735 . . . . . . . . . . . 12  |-  ( ( ( a  e.  om  /\ 
-.  a  e.  ran  E )  /\  ( b  e.  om  /\  a  =  suc  b ) )  ->  a  =  suc  b )
6160eqcomd 2443 . . . . . . . . . . 11  |-  ( ( ( a  e.  om  /\ 
-.  a  e.  ran  E )  /\  ( b  e.  om  /\  a  =  suc  b ) )  ->  suc  b  =  a )
6259, 61jca 520 . . . . . . . . . 10  |-  ( ( ( a  e.  om  /\ 
-.  a  e.  ran  E )  /\  ( b  e.  om  /\  a  =  suc  b ) )  ->  ( b  e. 
ran  E  /\  suc  b  =  a ) )
6362ex 425 . . . . . . . . 9  |-  ( ( a  e.  om  /\  -.  a  e.  ran  E )  ->  ( (
b  e.  om  /\  a  =  suc  b )  ->  ( b  e. 
ran  E  /\  suc  b  =  a ) ) )
6463reximdv2 2817 . . . . . . . 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 15 . . . . . . 7  |-  ( ( a  e.  om  /\  -.  a  e.  ran  E )  ->  E. b  e.  ran  E  suc  b  =  a )
6630, 65impbii 182 . . . . . 6  |-  ( E. b  e.  ran  E  suc  b  =  a  <->  ( a  e.  om  /\  -.  a  e.  ran  E ) )
6716, 66bitri 242 . . . . 5  |-  ( E. b  e.  ran  E
( S `  b
)  =  a  <->  ( a  e.  om  /\  -.  a  e.  ran  E ) )
68 f1fn 5642 . . . . . . 7  |-  ( S : On -1-1-> On  ->  S  Fn  On )
692, 68ax-mp 8 . . . . . 6  |-  S  Fn  On
70 fvelimab 5784 . . . . . 6  |-  ( ( S  Fn  On  /\  ran  E  C_  On )  ->  ( a  e.  ( S " ran  E
)  <->  E. b  e.  ran  E ( S `  b
)  =  a ) )
7169, 9, 70mp2an 655 . . . . 5  |-  ( a  e.  ( S " ran  E )  <->  E. b  e.  ran  E ( S `
 b )  =  a )
72 eldif 3332 . . . . 5  |-  ( a  e.  ( om  \  ran  E )  <->  ( a  e. 
om  /\  -.  a  e.  ran  E ) )
7367, 71, 723bitr4i 270 . . . 4  |-  ( a  e.  ( S " ran  E )  <->  a  e.  ( om  \  ran  E
) )
7473eqriv 2435 . . 3  |-  ( S
" ran  E )  =  ( om  \  ran  E )
75 f1oeq3 5669 . . 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 201 1  |-  ( S  |`  ran  E ) : ran  E -1-1-onto-> ( om  \  ran  E )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708    \ cdif 3319    C_ wss 3322   (/)c0 3630    e. cmpt 4268   Oncon0 4583   suc csuc 4585   omcom 4847   dom cdm 4880   ran crn 4881    |` cres 4882   "cima 4883   Fun wfun 5450    Fn wfn 5451   -->wf 5452   -1-1->wf1 5453   -1-1-onto->wf1o 5455   ` cfv 5456  (class class class)co 6083   2oc2o 6720    .o comu 6724
This theorem is referenced by:  fin1a2lem7  8288
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-omul 6731
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