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Theorem rankr1id 9280
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 ∈ dom 𝑅1 ↔ (rank‘(𝑅1𝐴)) = 𝐴)

Proof of Theorem rankr1id
StepHypRef Expression
1 ssid 3988 . . . 4 (𝑅1𝐴) ⊆ (𝑅1𝐴)
2 fvex 6677 . . . . . . . 8 (𝑅1𝐴) ∈ V
32pwid 4556 . . . . . . 7 (𝑅1𝐴) ∈ 𝒫 (𝑅1𝐴)
4 r1sucg 9187 . . . . . . 7 (𝐴 ∈ dom 𝑅1 → (𝑅1‘suc 𝐴) = 𝒫 (𝑅1𝐴))
53, 4eleqtrrid 2920 . . . . . 6 (𝐴 ∈ dom 𝑅1 → (𝑅1𝐴) ∈ (𝑅1‘suc 𝐴))
6 r1elwf 9214 . . . . . 6 ((𝑅1𝐴) ∈ (𝑅1‘suc 𝐴) → (𝑅1𝐴) ∈ (𝑅1 “ On))
75, 6syl 17 . . . . 5 (𝐴 ∈ dom 𝑅1 → (𝑅1𝐴) ∈ (𝑅1 “ On))
8 rankr1bg 9221 . . . . 5 (((𝑅1𝐴) ∈ (𝑅1 “ On) ∧ 𝐴 ∈ dom 𝑅1) → ((𝑅1𝐴) ⊆ (𝑅1𝐴) ↔ (rank‘(𝑅1𝐴)) ⊆ 𝐴))
97, 8mpancom 684 . . . 4 (𝐴 ∈ dom 𝑅1 → ((𝑅1𝐴) ⊆ (𝑅1𝐴) ↔ (rank‘(𝑅1𝐴)) ⊆ 𝐴))
101, 9mpbii 234 . . 3 (𝐴 ∈ dom 𝑅1 → (rank‘(𝑅1𝐴)) ⊆ 𝐴)
11 rankonid 9247 . . . . 5 (𝐴 ∈ dom 𝑅1 ↔ (rank‘𝐴) = 𝐴)
1211biimpi 217 . . . 4 (𝐴 ∈ dom 𝑅1 → (rank‘𝐴) = 𝐴)
13 onssr1 9249 . . . . 5 (𝐴 ∈ dom 𝑅1𝐴 ⊆ (𝑅1𝐴))
14 rankssb 9266 . . . . 5 ((𝑅1𝐴) ∈ (𝑅1 “ On) → (𝐴 ⊆ (𝑅1𝐴) → (rank‘𝐴) ⊆ (rank‘(𝑅1𝐴))))
157, 13, 14sylc 65 . . . 4 (𝐴 ∈ dom 𝑅1 → (rank‘𝐴) ⊆ (rank‘(𝑅1𝐴)))
1612, 15eqsstrrd 4005 . . 3 (𝐴 ∈ dom 𝑅1𝐴 ⊆ (rank‘(𝑅1𝐴)))
1710, 16eqssd 3983 . 2 (𝐴 ∈ dom 𝑅1 → (rank‘(𝑅1𝐴)) = 𝐴)
18 id 22 . . 3 ((rank‘(𝑅1𝐴)) = 𝐴 → (rank‘(𝑅1𝐴)) = 𝐴)
19 rankdmr1 9219 . . 3 (rank‘(𝑅1𝐴)) ∈ dom 𝑅1
2018, 19syl6eqelr 2922 . 2 ((rank‘(𝑅1𝐴)) = 𝐴𝐴 ∈ dom 𝑅1)
2117, 20impbii 210 1 (𝐴 ∈ dom 𝑅1 ↔ (rank‘(𝑅1𝐴)) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 207   = wceq 1528  wcel 2105  wss 3935  𝒫 cpw 4537   cuni 4832  dom cdm 5549  cima 5552  Oncon0 6185  suc csuc 6187  cfv 6349  𝑅1cr1 9180  rankcrnk 9181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4833  df-int 4870  df-iun 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-om 7569  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-r1 9182  df-rank 9183
This theorem is referenced by:  rankuni  9281
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