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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 3988 | . . . 4 ⊢ (𝑅1‘𝐴) ⊆ (𝑅1‘𝐴) | |
2 | fvex 6682 | . . . . . . . 8 ⊢ (𝑅1‘𝐴) ∈ V | |
3 | 2 | pwid 4562 | . . . . . . 7 ⊢ (𝑅1‘𝐴) ∈ 𝒫 (𝑅1‘𝐴) |
4 | r1sucg 9197 | . . . . . . 7 ⊢ (𝐴 ∈ dom 𝑅1 → (𝑅1‘suc 𝐴) = 𝒫 (𝑅1‘𝐴)) | |
5 | 3, 4 | eleqtrrid 2920 | . . . . . 6 ⊢ (𝐴 ∈ dom 𝑅1 → (𝑅1‘𝐴) ∈ (𝑅1‘suc 𝐴)) |
6 | r1elwf 9224 | . . . . . 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 9231 | . . . . 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 235 | . . 3 ⊢ (𝐴 ∈ dom 𝑅1 → (rank‘(𝑅1‘𝐴)) ⊆ 𝐴) |
11 | rankonid 9257 | . . . . 5 ⊢ (𝐴 ∈ dom 𝑅1 ↔ (rank‘𝐴) = 𝐴) | |
12 | 11 | biimpi 218 | . . . 4 ⊢ (𝐴 ∈ dom 𝑅1 → (rank‘𝐴) = 𝐴) |
13 | onssr1 9259 | . . . . 5 ⊢ (𝐴 ∈ dom 𝑅1 → 𝐴 ⊆ (𝑅1‘𝐴)) | |
14 | rankssb 9276 | . . . . 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 4005 | . . 3 ⊢ (𝐴 ∈ dom 𝑅1 → 𝐴 ⊆ (rank‘(𝑅1‘𝐴))) |
17 | 10, 16 | eqssd 3983 | . 2 ⊢ (𝐴 ∈ dom 𝑅1 → (rank‘(𝑅1‘𝐴)) = 𝐴) |
18 | id 22 | . . 3 ⊢ ((rank‘(𝑅1‘𝐴)) = 𝐴 → (rank‘(𝑅1‘𝐴)) = 𝐴) | |
19 | rankdmr1 9229 | . . 3 ⊢ (rank‘(𝑅1‘𝐴)) ∈ dom 𝑅1 | |
20 | 18, 19 | eqeltrrdi 2922 | . 2 ⊢ ((rank‘(𝑅1‘𝐴)) = 𝐴 → 𝐴 ∈ dom 𝑅1) |
21 | 17, 20 | impbii 211 | 1 ⊢ (𝐴 ∈ dom 𝑅1 ↔ (rank‘(𝑅1‘𝐴)) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 𝒫 cpw 4538 ∪ cuni 4837 dom cdm 5554 “ cima 5557 Oncon0 6190 suc csuc 6192 ‘cfv 6354 𝑅1cr1 9190 rankcrnk 9191 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-om 7580 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-r1 9192 df-rank 9193 |
This theorem is referenced by: rankuni 9291 |
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