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Mirrors > Home > MPE Home > Th. List > rankr1id | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
rankr1id | ⊢ (𝐴 ∈ dom 𝑅1 ↔ (rank‘(𝑅1‘𝐴)) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3964 | . . . 4 ⊢ (𝑅1‘𝐴) ⊆ (𝑅1‘𝐴) | |
2 | fvex 6852 | . . . . . . . 8 ⊢ (𝑅1‘𝐴) ∈ V | |
3 | 2 | pwid 4580 | . . . . . . 7 ⊢ (𝑅1‘𝐴) ∈ 𝒫 (𝑅1‘𝐴) |
4 | r1sucg 9701 | . . . . . . 7 ⊢ (𝐴 ∈ dom 𝑅1 → (𝑅1‘suc 𝐴) = 𝒫 (𝑅1‘𝐴)) | |
5 | 3, 4 | eleqtrrid 2845 | . . . . . 6 ⊢ (𝐴 ∈ dom 𝑅1 → (𝑅1‘𝐴) ∈ (𝑅1‘suc 𝐴)) |
6 | r1elwf 9728 | . . . . . 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 9735 | . . . . 5 ⊢ (((𝑅1‘𝐴) ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ∈ dom 𝑅1) → ((𝑅1‘𝐴) ⊆ (𝑅1‘𝐴) ↔ (rank‘(𝑅1‘𝐴)) ⊆ 𝐴)) | |
9 | 7, 8 | mpancom 686 | . . . 4 ⊢ (𝐴 ∈ dom 𝑅1 → ((𝑅1‘𝐴) ⊆ (𝑅1‘𝐴) ↔ (rank‘(𝑅1‘𝐴)) ⊆ 𝐴)) |
10 | 1, 9 | mpbii 232 | . . 3 ⊢ (𝐴 ∈ dom 𝑅1 → (rank‘(𝑅1‘𝐴)) ⊆ 𝐴) |
11 | rankonid 9761 | . . . . 5 ⊢ (𝐴 ∈ dom 𝑅1 ↔ (rank‘𝐴) = 𝐴) | |
12 | 11 | biimpi 215 | . . . 4 ⊢ (𝐴 ∈ dom 𝑅1 → (rank‘𝐴) = 𝐴) |
13 | onssr1 9763 | . . . . 5 ⊢ (𝐴 ∈ dom 𝑅1 → 𝐴 ⊆ (𝑅1‘𝐴)) | |
14 | rankssb 9780 | . . . . 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 3981 | . . 3 ⊢ (𝐴 ∈ dom 𝑅1 → 𝐴 ⊆ (rank‘(𝑅1‘𝐴))) |
17 | 10, 16 | eqssd 3959 | . 2 ⊢ (𝐴 ∈ dom 𝑅1 → (rank‘(𝑅1‘𝐴)) = 𝐴) |
18 | id 22 | . . 3 ⊢ ((rank‘(𝑅1‘𝐴)) = 𝐴 → (rank‘(𝑅1‘𝐴)) = 𝐴) | |
19 | rankdmr1 9733 | . . 3 ⊢ (rank‘(𝑅1‘𝐴)) ∈ dom 𝑅1 | |
20 | 18, 19 | eqeltrrdi 2847 | . 2 ⊢ ((rank‘(𝑅1‘𝐴)) = 𝐴 → 𝐴 ∈ dom 𝑅1) |
21 | 17, 20 | impbii 208 | 1 ⊢ (𝐴 ∈ dom 𝑅1 ↔ (rank‘(𝑅1‘𝐴)) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1541 ∈ wcel 2106 ⊆ wss 3908 𝒫 cpw 4558 ∪ cuni 4863 dom cdm 5631 “ cima 5634 Oncon0 6315 suc csuc 6317 ‘cfv 6493 𝑅1cr1 9694 rankcrnk 9695 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7356 df-om 7799 df-2nd 7918 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-r1 9696 df-rank 9697 |
This theorem is referenced by: rankuni 9795 |
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