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Theorem rankr1id 8330
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 3450 . . . 4  |-  ( R1
`  A )  C_  ( R1 `  A )
2 fvex 5873 . . . . . . . 8  |-  ( R1
`  A )  e. 
_V
32pwid 3964 . . . . . . 7  |-  ( R1
`  A )  e. 
~P ( R1 `  A )
4 r1sucg 8237 . . . . . . 7  |-  ( A  e.  dom  R1  ->  ( R1 `  suc  A
)  =  ~P ( R1 `  A ) )
53, 4syl5eleqr 2535 . . . . . 6  |-  ( A  e.  dom  R1  ->  ( R1 `  A )  e.  ( R1 `  suc  A ) )
6 r1elwf 8264 . . . . . 6  |-  ( ( R1 `  A )  e.  ( R1 `  suc  A )  ->  ( R1 `  A )  e. 
U. ( R1 " On ) )
75, 6syl 17 . . . . 5  |-  ( A  e.  dom  R1  ->  ( R1 `  A )  e.  U. ( R1
" On ) )
8 rankr1bg 8271 . . . . 5  |-  ( ( ( R1 `  A
)  e.  U. ( R1 " On )  /\  A  e.  dom  R1 )  ->  ( ( R1
`  A )  C_  ( R1 `  A )  <-> 
( rank `  ( R1 `  A ) )  C_  A ) )
97, 8mpancom 674 . . . 4  |-  ( A  e.  dom  R1  ->  ( ( R1 `  A
)  C_  ( R1 `  A )  <->  ( rank `  ( R1 `  A
) )  C_  A
) )
101, 9mpbii 215 . . 3  |-  ( A  e.  dom  R1  ->  (
rank `  ( R1 `  A ) )  C_  A )
11 rankonid 8297 . . . . 5  |-  ( A  e.  dom  R1  <->  ( rank `  A )  =  A )
1211biimpi 198 . . . 4  |-  ( A  e.  dom  R1  ->  (
rank `  A )  =  A )
13 onssr1 8299 . . . . 5  |-  ( A  e.  dom  R1  ->  A 
C_  ( R1 `  A ) )
14 rankssb 8316 . . . . 5  |-  ( ( R1 `  A )  e.  U. ( R1
" On )  -> 
( A  C_  ( R1 `  A )  -> 
( rank `  A )  C_  ( rank `  ( R1 `  A ) ) ) )
157, 13, 14sylc 62 . . . 4  |-  ( A  e.  dom  R1  ->  (
rank `  A )  C_  ( rank `  ( R1 `  A ) ) )
1612, 15eqsstr3d 3466 . . 3  |-  ( A  e.  dom  R1  ->  A 
C_  ( rank `  ( R1 `  A ) ) )
1710, 16eqssd 3448 . 2  |-  ( A  e.  dom  R1  ->  (
rank `  ( R1 `  A ) )  =  A )
18 id 22 . . 3  |-  ( (
rank `  ( R1 `  A ) )  =  A  ->  ( rank `  ( R1 `  A
) )  =  A )
19 rankdmr1 8269 . . 3  |-  ( rank `  ( R1 `  A
) )  e.  dom  R1
2018, 19syl6eqelr 2537 . 2  |-  ( (
rank `  ( R1 `  A ) )  =  A  ->  A  e.  dom  R1 )
2117, 20impbii 191 1  |-  ( A  e.  dom  R1  <->  ( rank `  ( R1 `  A
) )  =  A )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    = wceq 1443    e. wcel 1886    C_ wss 3403   ~Pcpw 3950   U.cuni 4197   dom cdm 4833   "cima 4836   Oncon0 5422   suc csuc 5424   ` cfv 5581   R1cr1 8230   rankcrnk 8231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-om 6690  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-r1 8232  df-rank 8233
This theorem is referenced by:  rankuni  8331
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