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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 9837 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 9768 | . . . 4 ⊢ 𝑅1 Fn On | |
2 | 1 | fndmi 6653 | . . 3 ⊢ dom 𝑅1 = On |
3 | 2 | eleq2i 2824 | . 2 ⊢ (𝐵 ∈ dom 𝑅1 ↔ 𝐵 ∈ On) |
4 | rankr1b.1 | . . . 4 ⊢ 𝐴 ∈ V | |
5 | unir1 9814 | . . . 4 ⊢ ∪ (𝑅1 “ On) = V | |
6 | 4, 5 | eleqtrri 2831 | . . 3 ⊢ 𝐴 ∈ ∪ (𝑅1 “ On) |
7 | rankr1bg 9804 | . . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ⊆ (𝑅1‘𝐵) ↔ (rank‘𝐴) ⊆ 𝐵)) | |
8 | 6, 7 | mpan 687 | . 2 ⊢ (𝐵 ∈ dom 𝑅1 → (𝐴 ⊆ (𝑅1‘𝐵) ↔ (rank‘𝐴) ⊆ 𝐵)) |
9 | 3, 8 | sylbir 234 | 1 ⊢ (𝐵 ∈ On → (𝐴 ⊆ (𝑅1‘𝐵) ↔ (rank‘𝐴) ⊆ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2105 Vcvv 3473 ⊆ wss 3948 ∪ cuni 4908 dom cdm 5676 “ cima 5679 Oncon0 6364 ‘cfv 6543 𝑅1cr1 9763 rankcrnk 9764 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-reg 9593 ax-inf2 9642 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-om 7860 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-r1 9765 df-rank 9766 |
This theorem is referenced by: (None) |
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