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Theorem rankr1id 9290
 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 3938 . . . 4 (𝑅1𝐴) ⊆ (𝑅1𝐴)
2 fvex 6665 . . . . . . . 8 (𝑅1𝐴) ∈ V
32pwid 4523 . . . . . . 7 (𝑅1𝐴) ∈ 𝒫 (𝑅1𝐴)
4 r1sucg 9197 . . . . . . 7 (𝐴 ∈ dom 𝑅1 → (𝑅1‘suc 𝐴) = 𝒫 (𝑅1𝐴))
53, 4eleqtrrid 2897 . . . . . 6 (𝐴 ∈ dom 𝑅1 → (𝑅1𝐴) ∈ (𝑅1‘suc 𝐴))
6 r1elwf 9224 . . . . . 6 ((𝑅1𝐴) ∈ (𝑅1‘suc 𝐴) → (𝑅1𝐴) ∈ (𝑅1 “ On))
75, 6syl 17 . . . . 5 (𝐴 ∈ dom 𝑅1 → (𝑅1𝐴) ∈ (𝑅1 “ On))
8 rankr1bg 9231 . . . . 5 (((𝑅1𝐴) ∈ (𝑅1 “ On) ∧ 𝐴 ∈ dom 𝑅1) → ((𝑅1𝐴) ⊆ (𝑅1𝐴) ↔ (rank‘(𝑅1𝐴)) ⊆ 𝐴))
97, 8mpancom 687 . . . 4 (𝐴 ∈ dom 𝑅1 → ((𝑅1𝐴) ⊆ (𝑅1𝐴) ↔ (rank‘(𝑅1𝐴)) ⊆ 𝐴))
101, 9mpbii 236 . . 3 (𝐴 ∈ dom 𝑅1 → (rank‘(𝑅1𝐴)) ⊆ 𝐴)
11 rankonid 9257 . . . . 5 (𝐴 ∈ dom 𝑅1 ↔ (rank‘𝐴) = 𝐴)
1211biimpi 219 . . . 4 (𝐴 ∈ dom 𝑅1 → (rank‘𝐴) = 𝐴)
13 onssr1 9259 . . . . 5 (𝐴 ∈ dom 𝑅1𝐴 ⊆ (𝑅1𝐴))
14 rankssb 9276 . . . . 5 ((𝑅1𝐴) ∈ (𝑅1 “ On) → (𝐴 ⊆ (𝑅1𝐴) → (rank‘𝐴) ⊆ (rank‘(𝑅1𝐴))))
157, 13, 14sylc 65 . . . 4 (𝐴 ∈ dom 𝑅1 → (rank‘𝐴) ⊆ (rank‘(𝑅1𝐴)))
1612, 15eqsstrrd 3955 . . 3 (𝐴 ∈ dom 𝑅1𝐴 ⊆ (rank‘(𝑅1𝐴)))
1710, 16eqssd 3933 . 2 (𝐴 ∈ dom 𝑅1 → (rank‘(𝑅1𝐴)) = 𝐴)
18 id 22 . . 3 ((rank‘(𝑅1𝐴)) = 𝐴 → (rank‘(𝑅1𝐴)) = 𝐴)
19 rankdmr1 9229 . . 3 (rank‘(𝑅1𝐴)) ∈ dom 𝑅1
2018, 19eqeltrrdi 2899 . 2 ((rank‘(𝑅1𝐴)) = 𝐴𝐴 ∈ dom 𝑅1)
2117, 20impbii 212 1 (𝐴 ∈ dom 𝑅1 ↔ (rank‘(𝑅1𝐴)) = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   = wceq 1538   ∈ wcel 2111   ⊆ wss 3882  𝒫 cpw 4499  ∪ cuni 4803  dom cdm 5522   “ cima 5525  Oncon0 6164  suc csuc 6166  ‘cfv 6329  𝑅1cr1 9190  rankcrnk 9191 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7451 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3722  df-csb 3830  df-dif 3885  df-un 3887  df-in 3889  df-ss 3899  df-pss 3901  df-nul 4246  df-if 4428  df-pw 4501  df-sn 4528  df-pr 4530  df-tp 4532  df-op 4534  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5441  df-so 5442  df-fr 5481  df-we 5483  df-xp 5528  df-rel 5529  df-cnv 5530  df-co 5531  df-dm 5532  df-rn 5533  df-res 5534  df-ima 5535  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6288  df-fun 6331  df-fn 6332  df-f 6333  df-f1 6334  df-fo 6335  df-f1o 6336  df-fv 6337  df-om 7571  df-wrecs 7945  df-recs 8006  df-rdg 8044  df-r1 9192  df-rank 9193 This theorem is referenced by:  rankuni  9291
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