Theorem rankr1id 9758

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‘(𝑅1‘𝐴)) = 𝐴)

Proof of Theorem rankr1id

Step	Hyp	Ref	Expression
1		ssid 3958	. . . 4 ⊢ (𝑅1‘𝐴) ⊆ (𝑅1‘𝐴)
2		fvex 6835	. . . . . . . 8 ⊢ (𝑅1‘𝐴) ∈ V
3	2	pwid 4573	. . . . . . 7 ⊢ (𝑅1‘𝐴) ∈ 𝒫 (𝑅1‘𝐴)
4		r1sucg 9665	. . . . . . 7 ⊢ (𝐴 ∈ dom 𝑅1 → (𝑅1‘suc 𝐴) = 𝒫 (𝑅1‘𝐴))
5	3, 4	eleqtrrid 2835	. . . . . 6 ⊢ (𝐴 ∈ dom 𝑅1 → (𝑅1‘𝐴) ∈ (𝑅1‘suc 𝐴))
6		r1elwf 9692	. . . . . 6 ⊢ ((𝑅1‘𝐴) ∈ (𝑅1‘suc 𝐴) → (𝑅1‘𝐴) ∈ ∪ (𝑅1 “ On))
7	5, 6	syl 17	. . . . 5 ⊢ (𝐴 ∈ dom 𝑅1 → (𝑅1‘𝐴) ∈ ∪ (𝑅1 “ On))
8		rankr1bg 9699	. . . . 5 ⊢ (((𝑅1‘𝐴) ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ∈ dom 𝑅1) → ((𝑅1‘𝐴) ⊆ (𝑅1‘𝐴) ↔ (rank‘(𝑅1‘𝐴)) ⊆ 𝐴))
9	7, 8	mpancom 688	. . . 4 ⊢ (𝐴 ∈ dom 𝑅1 → ((𝑅1‘𝐴) ⊆ (𝑅1‘𝐴) ↔ (rank‘(𝑅1‘𝐴)) ⊆ 𝐴))
10	1, 9	mpbii 233	. . 3 ⊢ (𝐴 ∈ dom 𝑅1 → (rank‘(𝑅1‘𝐴)) ⊆ 𝐴)
11		rankonid 9725	. . . . 5 ⊢ (𝐴 ∈ dom 𝑅1 ↔ (rank‘𝐴) = 𝐴)
12	11	biimpi 216	. . . 4 ⊢ (𝐴 ∈ dom 𝑅1 → (rank‘𝐴) = 𝐴)
13		onssr1 9727	. . . . 5 ⊢ (𝐴 ∈ dom 𝑅1 → 𝐴 ⊆ (𝑅1‘𝐴))
14		rankssb 9744	. . . . 5 ⊢ ((𝑅1‘𝐴) ∈ ∪ (𝑅1 “ On) → (𝐴 ⊆ (𝑅1‘𝐴) → (rank‘𝐴) ⊆ (rank‘(𝑅1‘𝐴))))
15	7, 13, 14	sylc 65	. . . 4 ⊢ (𝐴 ∈ dom 𝑅1 → (rank‘𝐴) ⊆ (rank‘(𝑅1‘𝐴)))
16	12, 15	eqsstrrd 3971	. . 3 ⊢ (𝐴 ∈ dom 𝑅1 → 𝐴 ⊆ (rank‘(𝑅1‘𝐴)))
17	10, 16	eqssd 3953	. 2 ⊢ (𝐴 ∈ dom 𝑅1 → (rank‘(𝑅1‘𝐴)) = 𝐴)
18		id 22	. . 3 ⊢ ((rank‘(𝑅1‘𝐴)) = 𝐴 → (rank‘(𝑅1‘𝐴)) = 𝐴)
19		rankdmr1 9697	. . 3 ⊢ (rank‘(𝑅1‘𝐴)) ∈ dom 𝑅1
20	18, 19	eqeltrrdi 2837	. 2 ⊢ ((rank‘(𝑅1‘𝐴)) = 𝐴 → 𝐴 ∈ dom 𝑅1)
21	17, 20	impbii 209	1 ⊢ (𝐴 ∈ dom 𝑅1 ↔ (rank‘(𝑅1‘𝐴)) = 𝐴)

Syntax hints:   ↔ wb 206   = wceq 1540   ∈ wcel 2109   ⊆ wss 3903  𝒫 cpw 4551  ∪ cuni 4858  dom cdm 5619   “ cima 5622  Oncon0 6307  suc csuc 6309  ‘cfv 6482  𝑅₁cr1 9658  rankcrnk 9659

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-om 7800  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-r1 9660  df-rank 9661

This theorem is referenced by:  rankuni  9759