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Mirrors > Home > MPE Home > Th. List > rankr1b | Structured version Visualization version GIF version |
Description: A relationship between rank and 𝑅1. See rankr1a 9264 for the membership version. (Contributed by NM, 15-Sep-2006.) (Revised by Mario Carneiro, 17-Nov-2014.) |
Ref | Expression |
---|---|
rankr1b.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
rankr1b | ⊢ (𝐵 ∈ On → (𝐴 ⊆ (𝑅1‘𝐵) ↔ (rank‘𝐴) ⊆ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r1fnon 9195 | . . . 4 ⊢ 𝑅1 Fn On | |
2 | fndm 6454 | . . . 4 ⊢ (𝑅1 Fn On → dom 𝑅1 = On) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ dom 𝑅1 = On |
4 | 3 | eleq2i 2904 | . 2 ⊢ (𝐵 ∈ dom 𝑅1 ↔ 𝐵 ∈ On) |
5 | rankr1b.1 | . . . 4 ⊢ 𝐴 ∈ V | |
6 | unir1 9241 | . . . 4 ⊢ ∪ (𝑅1 “ On) = V | |
7 | 5, 6 | eleqtrri 2912 | . . 3 ⊢ 𝐴 ∈ ∪ (𝑅1 “ On) |
8 | rankr1bg 9231 | . . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ⊆ (𝑅1‘𝐵) ↔ (rank‘𝐴) ⊆ 𝐵)) | |
9 | 7, 8 | mpan 688 | . 2 ⊢ (𝐵 ∈ dom 𝑅1 → (𝐴 ⊆ (𝑅1‘𝐵) ↔ (rank‘𝐴) ⊆ 𝐵)) |
10 | 4, 9 | sylbir 237 | 1 ⊢ (𝐵 ∈ On → (𝐴 ⊆ (𝑅1‘𝐵) ↔ (rank‘𝐴) ⊆ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ⊆ wss 3935 ∪ cuni 4837 dom cdm 5554 “ cima 5557 Oncon0 6190 Fn wfn 6349 ‘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-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-reg 9055 ax-inf2 9103 |
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: (None) |
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