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Mirrors > Home > MPE Home > Th. List > rankr1bg | Structured version Visualization version GIF version |
Description: A relationship between rank and 𝑅1. See rankr1ag 9220 for the membership version. (Contributed by Mario Carneiro, 17-Nov-2014.) |
Ref | Expression |
---|---|
rankr1bg | ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ⊆ (𝑅1‘𝐵) ↔ (rank‘𝐴) ⊆ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r1funlim 9184 | . . . . 5 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1) | |
2 | 1 | simpri 486 | . . . 4 ⊢ Lim dom 𝑅1 |
3 | limsuc 7552 | . . . 4 ⊢ (Lim dom 𝑅1 → (𝐵 ∈ dom 𝑅1 ↔ suc 𝐵 ∈ dom 𝑅1)) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ (𝐵 ∈ dom 𝑅1 ↔ suc 𝐵 ∈ dom 𝑅1) |
5 | rankr1ag 9220 | . . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ suc 𝐵 ∈ dom 𝑅1) → (𝐴 ∈ (𝑅1‘suc 𝐵) ↔ (rank‘𝐴) ∈ suc 𝐵)) | |
6 | 4, 5 | sylan2b 593 | . 2 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ∈ (𝑅1‘suc 𝐵) ↔ (rank‘𝐴) ∈ suc 𝐵)) |
7 | r1sucg 9187 | . . . . 5 ⊢ (𝐵 ∈ dom 𝑅1 → (𝑅1‘suc 𝐵) = 𝒫 (𝑅1‘𝐵)) | |
8 | 7 | adantl 482 | . . . 4 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝑅1‘suc 𝐵) = 𝒫 (𝑅1‘𝐵)) |
9 | 8 | eleq2d 2898 | . . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ∈ (𝑅1‘suc 𝐵) ↔ 𝐴 ∈ 𝒫 (𝑅1‘𝐵))) |
10 | fvex 6677 | . . . 4 ⊢ (𝑅1‘𝐵) ∈ V | |
11 | 10 | elpw2 5240 | . . 3 ⊢ (𝐴 ∈ 𝒫 (𝑅1‘𝐵) ↔ 𝐴 ⊆ (𝑅1‘𝐵)) |
12 | 9, 11 | syl6rbb 289 | . 2 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ⊆ (𝑅1‘𝐵) ↔ 𝐴 ∈ (𝑅1‘suc 𝐵))) |
13 | rankon 9213 | . . 3 ⊢ (rank‘𝐴) ∈ On | |
14 | limord 6244 | . . . . . 6 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1) | |
15 | 2, 14 | ax-mp 5 | . . . . 5 ⊢ Ord dom 𝑅1 |
16 | ordelon 6209 | . . . . 5 ⊢ ((Ord dom 𝑅1 ∧ 𝐵 ∈ dom 𝑅1) → 𝐵 ∈ On) | |
17 | 15, 16 | mpan 686 | . . . 4 ⊢ (𝐵 ∈ dom 𝑅1 → 𝐵 ∈ On) |
18 | 17 | adantl 482 | . . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → 𝐵 ∈ On) |
19 | onsssuc 6272 | . . 3 ⊢ (((rank‘𝐴) ∈ On ∧ 𝐵 ∈ On) → ((rank‘𝐴) ⊆ 𝐵 ↔ (rank‘𝐴) ∈ suc 𝐵)) | |
20 | 13, 18, 19 | sylancr 587 | . 2 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → ((rank‘𝐴) ⊆ 𝐵 ↔ (rank‘𝐴) ∈ suc 𝐵)) |
21 | 6, 12, 20 | 3bitr4d 312 | 1 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ⊆ (𝑅1‘𝐵) ↔ (rank‘𝐴) ⊆ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ⊆ wss 3935 𝒫 cpw 4537 ∪ cuni 4832 dom cdm 5549 “ cima 5552 Ord word 6184 Oncon0 6185 Lim wlim 6186 suc csuc 6187 Fun wfun 6343 ‘cfv 6349 𝑅1cr1 9180 rankcrnk 9181 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 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 3497 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 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-int 4870 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-om 7569 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-r1 9182 df-rank 9183 |
This theorem is referenced by: r1rankidb 9222 rankval3b 9244 rankssb 9266 rankeq0b 9278 rankr1id 9280 rankr1b 9282 |
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