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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 3991 | . . . 4 ⊢ (𝑅1‘𝐴) ⊆ (𝑅1‘𝐴) | |
2 | fvex 6685 | . . . . . . . 8 ⊢ (𝑅1‘𝐴) ∈ V | |
3 | 2 | pwid 4565 | . . . . . . 7 ⊢ (𝑅1‘𝐴) ∈ 𝒫 (𝑅1‘𝐴) |
4 | r1sucg 9200 | . . . . . . 7 ⊢ (𝐴 ∈ dom 𝑅1 → (𝑅1‘suc 𝐴) = 𝒫 (𝑅1‘𝐴)) | |
5 | 3, 4 | eleqtrrid 2922 | . . . . . 6 ⊢ (𝐴 ∈ dom 𝑅1 → (𝑅1‘𝐴) ∈ (𝑅1‘suc 𝐴)) |
6 | r1elwf 9227 | . . . . . 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 9234 | . . . . 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 9260 | . . . . 5 ⊢ (𝐴 ∈ dom 𝑅1 ↔ (rank‘𝐴) = 𝐴) | |
12 | 11 | biimpi 218 | . . . 4 ⊢ (𝐴 ∈ dom 𝑅1 → (rank‘𝐴) = 𝐴) |
13 | onssr1 9262 | . . . . 5 ⊢ (𝐴 ∈ dom 𝑅1 → 𝐴 ⊆ (𝑅1‘𝐴)) | |
14 | rankssb 9279 | . . . . 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 4008 | . . 3 ⊢ (𝐴 ∈ dom 𝑅1 → 𝐴 ⊆ (rank‘(𝑅1‘𝐴))) |
17 | 10, 16 | eqssd 3986 | . 2 ⊢ (𝐴 ∈ dom 𝑅1 → (rank‘(𝑅1‘𝐴)) = 𝐴) |
18 | id 22 | . . 3 ⊢ ((rank‘(𝑅1‘𝐴)) = 𝐴 → (rank‘(𝑅1‘𝐴)) = 𝐴) | |
19 | rankdmr1 9232 | . . 3 ⊢ (rank‘(𝑅1‘𝐴)) ∈ dom 𝑅1 | |
20 | 18, 19 | eqeltrrdi 2924 | . 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 1537 ∈ wcel 2114 ⊆ wss 3938 𝒫 cpw 4541 ∪ cuni 4840 dom cdm 5557 “ cima 5560 Oncon0 6193 suc csuc 6195 ‘cfv 6357 𝑅1cr1 9193 rankcrnk 9194 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-r1 9195 df-rank 9196 |
This theorem is referenced by: rankuni 9294 |
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