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Theorem rankr1id 7721
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 3310 . . . 4  |-  ( R1
`  A )  C_  ( R1 `  A )
2 fvex 5682 . . . . . . . 8  |-  ( R1
`  A )  e. 
_V
32pwid 3755 . . . . . . 7  |-  ( R1
`  A )  e. 
~P ( R1 `  A )
4 r1sucg 7628 . . . . . . 7  |-  ( A  e.  dom  R1  ->  ( R1 `  suc  A
)  =  ~P ( R1 `  A ) )
53, 4syl5eleqr 2474 . . . . . 6  |-  ( A  e.  dom  R1  ->  ( R1 `  A )  e.  ( R1 `  suc  A ) )
6 r1elwf 7655 . . . . . 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 7662 . . . . 5  |-  ( ( ( R1 `  A
)  e.  U. ( R1 " On )  /\  A  e.  dom  R1 )  ->  ( ( R1
`  A )  C_  ( R1 `  A )  <-> 
( rank `  ( R1 `  A ) )  C_  A ) )
97, 8mpancom 651 . . . 4  |-  ( A  e.  dom  R1  ->  ( ( R1 `  A
)  C_  ( R1 `  A )  <->  ( rank `  ( R1 `  A
) )  C_  A
) )
101, 9mpbii 203 . . 3  |-  ( A  e.  dom  R1  ->  (
rank `  ( R1 `  A ) )  C_  A )
11 rankonid 7688 . . . . 5  |-  ( A  e.  dom  R1  <->  ( rank `  A )  =  A )
1211biimpi 187 . . . 4  |-  ( A  e.  dom  R1  ->  (
rank `  A )  =  A )
13 onssr1 7690 . . . . 5  |-  ( A  e.  dom  R1  ->  A 
C_  ( R1 `  A ) )
14 rankssb 7707 . . . . 5  |-  ( ( R1 `  A )  e.  U. ( R1
" On )  -> 
( A  C_  ( R1 `  A )  -> 
( rank `  A )  C_  ( rank `  ( R1 `  A ) ) ) )
157, 13, 14sylc 58 . . . 4  |-  ( A  e.  dom  R1  ->  (
rank `  A )  C_  ( rank `  ( R1 `  A ) ) )
1612, 15eqsstr3d 3326 . . 3  |-  ( A  e.  dom  R1  ->  A 
C_  ( rank `  ( R1 `  A ) ) )
1710, 16eqssd 3308 . 2  |-  ( A  e.  dom  R1  ->  (
rank `  ( R1 `  A ) )  =  A )
18 id 20 . . 3  |-  ( (
rank `  ( R1 `  A ) )  =  A  ->  ( rank `  ( R1 `  A
) )  =  A )
19 rankdmr1 7660 . . 3  |-  ( rank `  ( R1 `  A
) )  e.  dom  R1
2018, 19syl6eqelr 2476 . 2  |-  ( (
rank `  ( R1 `  A ) )  =  A  ->  A  e.  dom  R1 )
2117, 20impbii 181 1  |-  ( A  e.  dom  R1  <->  ( rank `  ( R1 `  A
) )  =  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1649    e. wcel 1717    C_ wss 3263   ~Pcpw 3742   U.cuni 3957   Oncon0 4522   suc csuc 4524   dom cdm 4818   "cima 4821   ` cfv 5394   R1cr1 7621   rankcrnk 7622
This theorem is referenced by:  rankuni  7722
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-recs 6569  df-rdg 6604  df-r1 7623  df-rank 7624
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