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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 18994 | . . 3 ⊢ ∥r = (𝑤 ∈ V ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ (Base‘𝑤) ∧ ∃𝑧 ∈ (Base‘𝑤)(𝑧(.r‘𝑤)𝑥) = 𝑦)}) | |
2 | 1 | relmptopab 7142 | . 2 ⊢ Rel (∥r‘𝑅) |
3 | reldvdsr.1 | . . 3 ⊢ ∥ = (∥r‘𝑅) | |
4 | 3 | releqi 5436 | . 2 ⊢ (Rel ∥ ↔ Rel (∥r‘𝑅)) |
5 | 2, 4 | mpbir 223 | 1 ⊢ Rel ∥ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 386 = wceq 1658 ∈ wcel 2166 ∃wrex 3117 Vcvv 3413 Rel wrel 5346 ‘cfv 6122 (class class class)co 6904 Basecbs 16221 .rcmulr 16305 ∥rcdsr 18991 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ral 3121 df-rex 3122 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-nul 4144 df-if 4306 df-sn 4397 df-pr 4399 df-op 4403 df-uni 4658 df-br 4873 df-opab 4935 df-mpt 4952 df-id 5249 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-iota 6085 df-fun 6124 df-fv 6130 df-dvdsr 18994 |
This theorem is referenced by: dvdsr 18999 isunit 19010 subrgdvds 19149 |
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