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Theorem rankr1id 8061
Description: The rank of the hierarchy of an ordinal number is itself. (Contributed by NM, 14-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)
Assertion
Ref Expression
rankr1id  |-  ( A  e.  dom  R1  <->  ( rank `  ( R1 `  A
) )  =  A )

Proof of Theorem rankr1id
StepHypRef Expression
1 ssid 3370 . . . 4  |-  ( R1
`  A )  C_  ( R1 `  A )
2 fvex 5696 . . . . . . . 8  |-  ( R1
`  A )  e. 
_V
32pwid 3869 . . . . . . 7  |-  ( R1
`  A )  e. 
~P ( R1 `  A )
4 r1sucg 7968 . . . . . . 7  |-  ( A  e.  dom  R1  ->  ( R1 `  suc  A
)  =  ~P ( R1 `  A ) )
53, 4syl5eleqr 2525 . . . . . 6  |-  ( A  e.  dom  R1  ->  ( R1 `  A )  e.  ( R1 `  suc  A ) )
6 r1elwf 7995 . . . . . 6  |-  ( ( R1 `  A )  e.  ( R1 `  suc  A )  ->  ( R1 `  A )  e. 
U. ( R1 " On ) )
75, 6syl 16 . . . . 5  |-  ( A  e.  dom  R1  ->  ( R1 `  A )  e.  U. ( R1
" On ) )
8 rankr1bg 8002 . . . . 5  |-  ( ( ( R1 `  A
)  e.  U. ( R1 " On )  /\  A  e.  dom  R1 )  ->  ( ( R1
`  A )  C_  ( R1 `  A )  <-> 
( rank `  ( R1 `  A ) )  C_  A ) )
97, 8mpancom 669 . . . 4  |-  ( A  e.  dom  R1  ->  ( ( R1 `  A
)  C_  ( R1 `  A )  <->  ( rank `  ( R1 `  A
) )  C_  A
) )
101, 9mpbii 211 . . 3  |-  ( A  e.  dom  R1  ->  (
rank `  ( R1 `  A ) )  C_  A )
11 rankonid 8028 . . . . 5  |-  ( A  e.  dom  R1  <->  ( rank `  A )  =  A )
1211biimpi 194 . . . 4  |-  ( A  e.  dom  R1  ->  (
rank `  A )  =  A )
13 onssr1 8030 . . . . 5  |-  ( A  e.  dom  R1  ->  A 
C_  ( R1 `  A ) )
14 rankssb 8047 . . . . 5  |-  ( ( R1 `  A )  e.  U. ( R1
" On )  -> 
( A  C_  ( R1 `  A )  -> 
( rank `  A )  C_  ( rank `  ( R1 `  A ) ) ) )
157, 13, 14sylc 60 . . . 4  |-  ( A  e.  dom  R1  ->  (
rank `  A )  C_  ( rank `  ( R1 `  A ) ) )
1612, 15eqsstr3d 3386 . . 3  |-  ( A  e.  dom  R1  ->  A 
C_  ( rank `  ( R1 `  A ) ) )
1710, 16eqssd 3368 . 2  |-  ( A  e.  dom  R1  ->  (
rank `  ( R1 `  A ) )  =  A )
18 id 22 . . 3  |-  ( (
rank `  ( R1 `  A ) )  =  A  ->  ( rank `  ( R1 `  A
) )  =  A )
19 rankdmr1 8000 . . 3  |-  ( rank `  ( R1 `  A
) )  e.  dom  R1
2018, 19syl6eqelr 2527 . 2  |-  ( (
rank `  ( R1 `  A ) )  =  A  ->  A  e.  dom  R1 )
2117, 20impbii 188 1  |-  ( A  e.  dom  R1  <->  ( rank `  ( R1 `  A
) )  =  A )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1369    e. wcel 1756    C_ wss 3323   ~Pcpw 3855   U.cuni 4086   Oncon0 4714   suc csuc 4716   dom cdm 4835   "cima 4838   ` cfv 5413   R1cr1 7961   rankcrnk 7962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-om 6472  df-recs 6824  df-rdg 6858  df-r1 7963  df-rank 7964
This theorem is referenced by:  rankuni  8062
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