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Theorem rankr1id 8608
 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 3587 . . . 4 (𝑅1𝐴) ⊆ (𝑅1𝐴)
2 fvex 6113 . . . . . . . 8 (𝑅1𝐴) ∈ V
32pwid 4122 . . . . . . 7 (𝑅1𝐴) ∈ 𝒫 (𝑅1𝐴)
4 r1sucg 8515 . . . . . . 7 (𝐴 ∈ dom 𝑅1 → (𝑅1‘suc 𝐴) = 𝒫 (𝑅1𝐴))
53, 4syl5eleqr 2695 . . . . . 6 (𝐴 ∈ dom 𝑅1 → (𝑅1𝐴) ∈ (𝑅1‘suc 𝐴))
6 r1elwf 8542 . . . . . 6 ((𝑅1𝐴) ∈ (𝑅1‘suc 𝐴) → (𝑅1𝐴) ∈ (𝑅1 “ On))
75, 6syl 17 . . . . 5 (𝐴 ∈ dom 𝑅1 → (𝑅1𝐴) ∈ (𝑅1 “ On))
8 rankr1bg 8549 . . . . 5 (((𝑅1𝐴) ∈ (𝑅1 “ On) ∧ 𝐴 ∈ dom 𝑅1) → ((𝑅1𝐴) ⊆ (𝑅1𝐴) ↔ (rank‘(𝑅1𝐴)) ⊆ 𝐴))
97, 8mpancom 700 . . . 4 (𝐴 ∈ dom 𝑅1 → ((𝑅1𝐴) ⊆ (𝑅1𝐴) ↔ (rank‘(𝑅1𝐴)) ⊆ 𝐴))
101, 9mpbii 222 . . 3 (𝐴 ∈ dom 𝑅1 → (rank‘(𝑅1𝐴)) ⊆ 𝐴)
11 rankonid 8575 . . . . 5 (𝐴 ∈ dom 𝑅1 ↔ (rank‘𝐴) = 𝐴)
1211biimpi 205 . . . 4 (𝐴 ∈ dom 𝑅1 → (rank‘𝐴) = 𝐴)
13 onssr1 8577 . . . . 5 (𝐴 ∈ dom 𝑅1𝐴 ⊆ (𝑅1𝐴))
14 rankssb 8594 . . . . 5 ((𝑅1𝐴) ∈ (𝑅1 “ On) → (𝐴 ⊆ (𝑅1𝐴) → (rank‘𝐴) ⊆ (rank‘(𝑅1𝐴))))
157, 13, 14sylc 63 . . . 4 (𝐴 ∈ dom 𝑅1 → (rank‘𝐴) ⊆ (rank‘(𝑅1𝐴)))
1612, 15eqsstr3d 3603 . . 3 (𝐴 ∈ dom 𝑅1𝐴 ⊆ (rank‘(𝑅1𝐴)))
1710, 16eqssd 3585 . 2 (𝐴 ∈ dom 𝑅1 → (rank‘(𝑅1𝐴)) = 𝐴)
18 id 22 . . 3 ((rank‘(𝑅1𝐴)) = 𝐴 → (rank‘(𝑅1𝐴)) = 𝐴)
19 rankdmr1 8547 . . 3 (rank‘(𝑅1𝐴)) ∈ dom 𝑅1
2018, 19syl6eqelr 2697 . 2 ((rank‘(𝑅1𝐴)) = 𝐴𝐴 ∈ dom 𝑅1)
2117, 20impbii 198 1 (𝐴 ∈ dom 𝑅1 ↔ (rank‘(𝑅1𝐴)) = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372  dom cdm 5038   “ cima 5041  Oncon0 5640  suc csuc 5642  ‘cfv 5804  𝑅1cr1 8508  rankcrnk 8509 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-r1 8510  df-rank 8511 This theorem is referenced by:  rankuni  8609
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