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Theorem rankr1id 7780
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 3359 . . . 4  |-  ( R1
`  A )  C_  ( R1 `  A )
2 fvex 5734 . . . . . . . 8  |-  ( R1
`  A )  e. 
_V
32pwid 3804 . . . . . . 7  |-  ( R1
`  A )  e. 
~P ( R1 `  A )
4 r1sucg 7687 . . . . . . 7  |-  ( A  e.  dom  R1  ->  ( R1 `  suc  A
)  =  ~P ( R1 `  A ) )
53, 4syl5eleqr 2522 . . . . . 6  |-  ( A  e.  dom  R1  ->  ( R1 `  A )  e.  ( R1 `  suc  A ) )
6 r1elwf 7714 . . . . . 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 7721 . . . . 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 7747 . . . . 5  |-  ( A  e.  dom  R1  <->  ( rank `  A )  =  A )
1211biimpi 187 . . . 4  |-  ( A  e.  dom  R1  ->  (
rank `  A )  =  A )
13 onssr1 7749 . . . . 5  |-  ( A  e.  dom  R1  ->  A 
C_  ( R1 `  A ) )
14 rankssb 7766 . . . . 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 3375 . . 3  |-  ( A  e.  dom  R1  ->  A 
C_  ( rank `  ( R1 `  A ) ) )
1710, 16eqssd 3357 . 2  |-  ( A  e.  dom  R1  ->  (
rank `  ( R1 `  A ) )  =  A )
18 id 20 . . 3  |-  ( (
rank `  ( R1 `  A ) )  =  A  ->  ( rank `  ( R1 `  A
) )  =  A )
19 rankdmr1 7719 . . 3  |-  ( rank `  ( R1 `  A
) )  e.  dom  R1
2018, 19syl6eqelr 2524 . 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 1652    e. wcel 1725    C_ wss 3312   ~Pcpw 3791   U.cuni 4007   Oncon0 4573   suc csuc 4575   dom cdm 4870   "cima 4873   ` cfv 5446   R1cr1 7680   rankcrnk 7681
This theorem is referenced by:  rankuni  7781
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-recs 6625  df-rdg 6660  df-r1 7682  df-rank 7683
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