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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 9304 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 9268 | . . . . 5 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1) | |
2 | 1 | simpri 489 | . . . 4 ⊢ Lim dom 𝑅1 |
3 | limsuc 7583 | . . . 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 9304 | . . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ suc 𝐵 ∈ dom 𝑅1) → (𝐴 ∈ (𝑅1‘suc 𝐵) ↔ (rank‘𝐴) ∈ suc 𝐵)) | |
6 | 4, 5 | sylan2b 597 | . 2 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ∈ (𝑅1‘suc 𝐵) ↔ (rank‘𝐴) ∈ suc 𝐵)) |
7 | r1sucg 9271 | . . . . 5 ⊢ (𝐵 ∈ dom 𝑅1 → (𝑅1‘suc 𝐵) = 𝒫 (𝑅1‘𝐵)) | |
8 | 7 | adantl 485 | . . . 4 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝑅1‘suc 𝐵) = 𝒫 (𝑅1‘𝐵)) |
9 | 8 | eleq2d 2818 | . . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ∈ (𝑅1‘suc 𝐵) ↔ 𝐴 ∈ 𝒫 (𝑅1‘𝐵))) |
10 | fvex 6687 | . . . 4 ⊢ (𝑅1‘𝐵) ∈ V | |
11 | 10 | elpw2 5213 | . . 3 ⊢ (𝐴 ∈ 𝒫 (𝑅1‘𝐵) ↔ 𝐴 ⊆ (𝑅1‘𝐵)) |
12 | 9, 11 | bitr2di 291 | . 2 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ⊆ (𝑅1‘𝐵) ↔ 𝐴 ∈ (𝑅1‘suc 𝐵))) |
13 | rankon 9297 | . . 3 ⊢ (rank‘𝐴) ∈ On | |
14 | limord 6231 | . . . . . 6 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1) | |
15 | 2, 14 | ax-mp 5 | . . . . 5 ⊢ Ord dom 𝑅1 |
16 | ordelon 6196 | . . . . 5 ⊢ ((Ord dom 𝑅1 ∧ 𝐵 ∈ dom 𝑅1) → 𝐵 ∈ On) | |
17 | 15, 16 | mpan 690 | . . . 4 ⊢ (𝐵 ∈ dom 𝑅1 → 𝐵 ∈ On) |
18 | 17 | adantl 485 | . . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → 𝐵 ∈ On) |
19 | onsssuc 6259 | . . 3 ⊢ (((rank‘𝐴) ∈ On ∧ 𝐵 ∈ On) → ((rank‘𝐴) ⊆ 𝐵 ↔ (rank‘𝐴) ∈ suc 𝐵)) | |
20 | 13, 18, 19 | sylancr 590 | . 2 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → ((rank‘𝐴) ⊆ 𝐵 ↔ (rank‘𝐴) ∈ suc 𝐵)) |
21 | 6, 12, 20 | 3bitr4d 314 | 1 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ⊆ (𝑅1‘𝐵) ↔ (rank‘𝐴) ⊆ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ⊆ wss 3843 𝒫 cpw 4488 ∪ cuni 4796 dom cdm 5525 “ cima 5528 Ord word 6171 Oncon0 6172 Lim wlim 6173 suc csuc 6174 Fun wfun 6333 ‘cfv 6339 𝑅1cr1 9264 rankcrnk 9265 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-om 7600 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-r1 9266 df-rank 9267 |
This theorem is referenced by: r1rankidb 9306 rankval3b 9328 rankssb 9350 rankeq0b 9362 rankr1id 9364 rankr1b 9366 |
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