MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rankr1id Structured version   Visualization version   GIF version

Theorem rankr1id 9902
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 4006 . . . 4 (𝑅1𝐴) ⊆ (𝑅1𝐴)
2 fvex 6919 . . . . . . . 8 (𝑅1𝐴) ∈ V
32pwid 4622 . . . . . . 7 (𝑅1𝐴) ∈ 𝒫 (𝑅1𝐴)
4 r1sucg 9809 . . . . . . 7 (𝐴 ∈ dom 𝑅1 → (𝑅1‘suc 𝐴) = 𝒫 (𝑅1𝐴))
53, 4eleqtrrid 2848 . . . . . 6 (𝐴 ∈ dom 𝑅1 → (𝑅1𝐴) ∈ (𝑅1‘suc 𝐴))
6 r1elwf 9836 . . . . . 6 ((𝑅1𝐴) ∈ (𝑅1‘suc 𝐴) → (𝑅1𝐴) ∈ (𝑅1 “ On))
75, 6syl 17 . . . . 5 (𝐴 ∈ dom 𝑅1 → (𝑅1𝐴) ∈ (𝑅1 “ On))
8 rankr1bg 9843 . . . . 5 (((𝑅1𝐴) ∈ (𝑅1 “ On) ∧ 𝐴 ∈ dom 𝑅1) → ((𝑅1𝐴) ⊆ (𝑅1𝐴) ↔ (rank‘(𝑅1𝐴)) ⊆ 𝐴))
97, 8mpancom 688 . . . 4 (𝐴 ∈ dom 𝑅1 → ((𝑅1𝐴) ⊆ (𝑅1𝐴) ↔ (rank‘(𝑅1𝐴)) ⊆ 𝐴))
101, 9mpbii 233 . . 3 (𝐴 ∈ dom 𝑅1 → (rank‘(𝑅1𝐴)) ⊆ 𝐴)
11 rankonid 9869 . . . . 5 (𝐴 ∈ dom 𝑅1 ↔ (rank‘𝐴) = 𝐴)
1211biimpi 216 . . . 4 (𝐴 ∈ dom 𝑅1 → (rank‘𝐴) = 𝐴)
13 onssr1 9871 . . . . 5 (𝐴 ∈ dom 𝑅1𝐴 ⊆ (𝑅1𝐴))
14 rankssb 9888 . . . . 5 ((𝑅1𝐴) ∈ (𝑅1 “ On) → (𝐴 ⊆ (𝑅1𝐴) → (rank‘𝐴) ⊆ (rank‘(𝑅1𝐴))))
157, 13, 14sylc 65 . . . 4 (𝐴 ∈ dom 𝑅1 → (rank‘𝐴) ⊆ (rank‘(𝑅1𝐴)))
1612, 15eqsstrrd 4019 . . 3 (𝐴 ∈ dom 𝑅1𝐴 ⊆ (rank‘(𝑅1𝐴)))
1710, 16eqssd 4001 . 2 (𝐴 ∈ dom 𝑅1 → (rank‘(𝑅1𝐴)) = 𝐴)
18 id 22 . . 3 ((rank‘(𝑅1𝐴)) = 𝐴 → (rank‘(𝑅1𝐴)) = 𝐴)
19 rankdmr1 9841 . . 3 (rank‘(𝑅1𝐴)) ∈ dom 𝑅1
2018, 19eqeltrrdi 2850 . 2 ((rank‘(𝑅1𝐴)) = 𝐴𝐴 ∈ dom 𝑅1)
2117, 20impbii 209 1 (𝐴 ∈ dom 𝑅1 ↔ (rank‘(𝑅1𝐴)) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1540  wcel 2108  wss 3951  𝒫 cpw 4600   cuni 4907  dom cdm 5685  cima 5688  Oncon0 6384  suc csuc 6386  cfv 6561  𝑅1cr1 9802  rankcrnk 9803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-r1 9804  df-rank 9805
This theorem is referenced by:  rankuni  9903
  Copyright terms: Public domain W3C validator