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Theorem rankr1id 8685
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 3609 . . . 4 (𝑅1𝐴) ⊆ (𝑅1𝐴)
2 fvex 6168 . . . . . . . 8 (𝑅1𝐴) ∈ V
32pwid 4152 . . . . . . 7 (𝑅1𝐴) ∈ 𝒫 (𝑅1𝐴)
4 r1sucg 8592 . . . . . . 7 (𝐴 ∈ dom 𝑅1 → (𝑅1‘suc 𝐴) = 𝒫 (𝑅1𝐴))
53, 4syl5eleqr 2705 . . . . . 6 (𝐴 ∈ dom 𝑅1 → (𝑅1𝐴) ∈ (𝑅1‘suc 𝐴))
6 r1elwf 8619 . . . . . 6 ((𝑅1𝐴) ∈ (𝑅1‘suc 𝐴) → (𝑅1𝐴) ∈ (𝑅1 “ On))
75, 6syl 17 . . . . 5 (𝐴 ∈ dom 𝑅1 → (𝑅1𝐴) ∈ (𝑅1 “ On))
8 rankr1bg 8626 . . . . 5 (((𝑅1𝐴) ∈ (𝑅1 “ On) ∧ 𝐴 ∈ dom 𝑅1) → ((𝑅1𝐴) ⊆ (𝑅1𝐴) ↔ (rank‘(𝑅1𝐴)) ⊆ 𝐴))
97, 8mpancom 702 . . . 4 (𝐴 ∈ dom 𝑅1 → ((𝑅1𝐴) ⊆ (𝑅1𝐴) ↔ (rank‘(𝑅1𝐴)) ⊆ 𝐴))
101, 9mpbii 223 . . 3 (𝐴 ∈ dom 𝑅1 → (rank‘(𝑅1𝐴)) ⊆ 𝐴)
11 rankonid 8652 . . . . 5 (𝐴 ∈ dom 𝑅1 ↔ (rank‘𝐴) = 𝐴)
1211biimpi 206 . . . 4 (𝐴 ∈ dom 𝑅1 → (rank‘𝐴) = 𝐴)
13 onssr1 8654 . . . . 5 (𝐴 ∈ dom 𝑅1𝐴 ⊆ (𝑅1𝐴))
14 rankssb 8671 . . . . 5 ((𝑅1𝐴) ∈ (𝑅1 “ On) → (𝐴 ⊆ (𝑅1𝐴) → (rank‘𝐴) ⊆ (rank‘(𝑅1𝐴))))
157, 13, 14sylc 65 . . . 4 (𝐴 ∈ dom 𝑅1 → (rank‘𝐴) ⊆ (rank‘(𝑅1𝐴)))
1612, 15eqsstr3d 3625 . . 3 (𝐴 ∈ dom 𝑅1𝐴 ⊆ (rank‘(𝑅1𝐴)))
1710, 16eqssd 3605 . 2 (𝐴 ∈ dom 𝑅1 → (rank‘(𝑅1𝐴)) = 𝐴)
18 id 22 . . 3 ((rank‘(𝑅1𝐴)) = 𝐴 → (rank‘(𝑅1𝐴)) = 𝐴)
19 rankdmr1 8624 . . 3 (rank‘(𝑅1𝐴)) ∈ dom 𝑅1
2018, 19syl6eqelr 2707 . 2 ((rank‘(𝑅1𝐴)) = 𝐴𝐴 ∈ dom 𝑅1)
2117, 20impbii 199 1 (𝐴 ∈ dom 𝑅1 ↔ (rank‘(𝑅1𝐴)) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1480  wcel 1987  wss 3560  𝒫 cpw 4136   cuni 4409  dom cdm 5084  cima 5087  Oncon0 5692  suc csuc 5694  cfv 5857  𝑅1cr1 8585  rankcrnk 8586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-om 7028  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-r1 8587  df-rank 8588
This theorem is referenced by:  rankuni  8686
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