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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 8699 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 8630 | . . . 4 ⊢ 𝑅1 Fn On | |
2 | fndm 5990 | . . . 4 ⊢ (𝑅1 Fn On → dom 𝑅1 = On) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ dom 𝑅1 = On |
4 | 3 | eleq2i 2693 | . 2 ⊢ (𝐵 ∈ dom 𝑅1 ↔ 𝐵 ∈ On) |
5 | rankr1b.1 | . . . 4 ⊢ 𝐴 ∈ V | |
6 | unir1 8676 | . . . 4 ⊢ ∪ (𝑅1 “ On) = V | |
7 | 5, 6 | eleqtrri 2700 | . . 3 ⊢ 𝐴 ∈ ∪ (𝑅1 “ On) |
8 | rankr1bg 8666 | . . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ⊆ (𝑅1‘𝐵) ↔ (rank‘𝐴) ⊆ 𝐵)) | |
9 | 7, 8 | mpan 706 | . 2 ⊢ (𝐵 ∈ dom 𝑅1 → (𝐴 ⊆ (𝑅1‘𝐵) ↔ (rank‘𝐴) ⊆ 𝐵)) |
10 | 4, 9 | sylbir 225 | 1 ⊢ (𝐵 ∈ On → (𝐴 ⊆ (𝑅1‘𝐵) ↔ (rank‘𝐴) ⊆ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ⊆ wss 3574 ∪ cuni 4436 dom cdm 5114 “ cima 5117 Oncon0 5723 Fn wfn 5883 ‘cfv 5888 𝑅1cr1 8625 rankcrnk 8626 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-reg 8497 ax-inf2 8538 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-r1 8627 df-rank 8628 |
This theorem is referenced by: (None) |
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