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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 3587 | . . . 4 ⊢ (𝑅1‘𝐴) ⊆ (𝑅1‘𝐴) | |
2 | fvex 6113 | . . . . . . . 8 ⊢ (𝑅1‘𝐴) ∈ V | |
3 | 2 | pwid 4122 | . . . . . . 7 ⊢ (𝑅1‘𝐴) ∈ 𝒫 (𝑅1‘𝐴) |
4 | r1sucg 8515 | . . . . . . 7 ⊢ (𝐴 ∈ dom 𝑅1 → (𝑅1‘suc 𝐴) = 𝒫 (𝑅1‘𝐴)) | |
5 | 3, 4 | syl5eleqr 2695 | . . . . . 6 ⊢ (𝐴 ∈ dom 𝑅1 → (𝑅1‘𝐴) ∈ (𝑅1‘suc 𝐴)) |
6 | r1elwf 8542 | . . . . . 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 8549 | . . . . 5 ⊢ (((𝑅1‘𝐴) ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ∈ dom 𝑅1) → ((𝑅1‘𝐴) ⊆ (𝑅1‘𝐴) ↔ (rank‘(𝑅1‘𝐴)) ⊆ 𝐴)) | |
9 | 7, 8 | mpancom 700 | . . . 4 ⊢ (𝐴 ∈ dom 𝑅1 → ((𝑅1‘𝐴) ⊆ (𝑅1‘𝐴) ↔ (rank‘(𝑅1‘𝐴)) ⊆ 𝐴)) |
10 | 1, 9 | mpbii 222 | . . 3 ⊢ (𝐴 ∈ dom 𝑅1 → (rank‘(𝑅1‘𝐴)) ⊆ 𝐴) |
11 | rankonid 8575 | . . . . 5 ⊢ (𝐴 ∈ dom 𝑅1 ↔ (rank‘𝐴) = 𝐴) | |
12 | 11 | biimpi 205 | . . . 4 ⊢ (𝐴 ∈ dom 𝑅1 → (rank‘𝐴) = 𝐴) |
13 | onssr1 8577 | . . . . 5 ⊢ (𝐴 ∈ dom 𝑅1 → 𝐴 ⊆ (𝑅1‘𝐴)) | |
14 | rankssb 8594 | . . . . 5 ⊢ ((𝑅1‘𝐴) ∈ ∪ (𝑅1 “ On) → (𝐴 ⊆ (𝑅1‘𝐴) → (rank‘𝐴) ⊆ (rank‘(𝑅1‘𝐴)))) | |
15 | 7, 13, 14 | sylc 63 | . . . 4 ⊢ (𝐴 ∈ dom 𝑅1 → (rank‘𝐴) ⊆ (rank‘(𝑅1‘𝐴))) |
16 | 12, 15 | eqsstr3d 3603 | . . 3 ⊢ (𝐴 ∈ dom 𝑅1 → 𝐴 ⊆ (rank‘(𝑅1‘𝐴))) |
17 | 10, 16 | eqssd 3585 | . 2 ⊢ (𝐴 ∈ dom 𝑅1 → (rank‘(𝑅1‘𝐴)) = 𝐴) |
18 | id 22 | . . 3 ⊢ ((rank‘(𝑅1‘𝐴)) = 𝐴 → (rank‘(𝑅1‘𝐴)) = 𝐴) | |
19 | rankdmr1 8547 | . . 3 ⊢ (rank‘(𝑅1‘𝐴)) ∈ dom 𝑅1 | |
20 | 18, 19 | syl6eqelr 2697 | . 2 ⊢ ((rank‘(𝑅1‘𝐴)) = 𝐴 → 𝐴 ∈ dom 𝑅1) |
21 | 17, 20 | impbii 198 | 1 ⊢ (𝐴 ∈ dom 𝑅1 ↔ (rank‘(𝑅1‘𝐴)) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 𝒫 cpw 4108 ∪ cuni 4372 dom cdm 5038 “ cima 5041 Oncon0 5640 suc csuc 5642 ‘cfv 5804 𝑅1cr1 8508 rankcrnk 8509 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-r1 8510 df-rank 8511 |
This theorem is referenced by: rankuni 8609 |
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