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Theorem rankr1id 8297
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 3518 . . . 4  |-  ( R1
`  A )  C_  ( R1 `  A )
2 fvex 5882 . . . . . . . 8  |-  ( R1
`  A )  e. 
_V
32pwid 4029 . . . . . . 7  |-  ( R1
`  A )  e. 
~P ( R1 `  A )
4 r1sucg 8204 . . . . . . 7  |-  ( A  e.  dom  R1  ->  ( R1 `  suc  A
)  =  ~P ( R1 `  A ) )
53, 4syl5eleqr 2552 . . . . . 6  |-  ( A  e.  dom  R1  ->  ( R1 `  A )  e.  ( R1 `  suc  A ) )
6 r1elwf 8231 . . . . . 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 8238 . . . . 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 8264 . . . . 5  |-  ( A  e.  dom  R1  <->  ( rank `  A )  =  A )
1211biimpi 194 . . . 4  |-  ( A  e.  dom  R1  ->  (
rank `  A )  =  A )
13 onssr1 8266 . . . . 5  |-  ( A  e.  dom  R1  ->  A 
C_  ( R1 `  A ) )
14 rankssb 8283 . . . . 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 3534 . . 3  |-  ( A  e.  dom  R1  ->  A 
C_  ( rank `  ( R1 `  A ) ) )
1710, 16eqssd 3516 . 2  |-  ( A  e.  dom  R1  ->  (
rank `  ( R1 `  A ) )  =  A )
18 id 22 . . 3  |-  ( (
rank `  ( R1 `  A ) )  =  A  ->  ( rank `  ( R1 `  A
) )  =  A )
19 rankdmr1 8236 . . 3  |-  ( rank `  ( R1 `  A
) )  e.  dom  R1
2018, 19syl6eqelr 2554 . 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 1395    e. wcel 1819    C_ wss 3471   ~Pcpw 4015   U.cuni 4251   Oncon0 4887   suc csuc 4889   dom cdm 5008   "cima 5011   ` cfv 5594   R1cr1 8197   rankcrnk 8198
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-om 6700  df-recs 7060  df-rdg 7094  df-r1 8199  df-rank 8200
This theorem is referenced by:  rankuni  8298
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