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Mirrors > Home > MPE Home > Th. List > reldvdsr | Structured version Visualization version GIF version |
Description: The divides relation is a relation. (Contributed by Mario Carneiro, 1-Dec-2014.) |
Ref | Expression |
---|---|
reldvdsr.1 | ⊢ ∥ = (∥r‘𝑅) |
Ref | Expression |
---|---|
reldvdsr | ⊢ Rel ∥ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dvdsr 19393 | . . 3 ⊢ ∥r = (𝑤 ∈ V ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ (Base‘𝑤) ∧ ∃𝑧 ∈ (Base‘𝑤)(𝑧(.r‘𝑤)𝑥) = 𝑦)}) | |
2 | 1 | relmptopab 7397 | . 2 ⊢ Rel (∥r‘𝑅) |
3 | reldvdsr.1 | . . 3 ⊢ ∥ = (∥r‘𝑅) | |
4 | 3 | releqi 5654 | . 2 ⊢ (Rel ∥ ↔ Rel (∥r‘𝑅)) |
5 | 2, 4 | mpbir 233 | 1 ⊢ Rel ∥ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3141 Vcvv 3496 Rel wrel 5562 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 .rcmulr 16568 ∥rcdsr 19390 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fv 6365 df-dvdsr 19393 |
This theorem is referenced by: dvdsr 19398 isunit 19409 subrgdvds 19551 |
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