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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 19383 | . . 3 ⊢ ∥r = (𝑤 ∈ V ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ (Base‘𝑤) ∧ ∃𝑧 ∈ (Base‘𝑤)(𝑧(.r‘𝑤)𝑥) = 𝑦)}) | |
2 | 1 | relmptopab 7387 | . 2 ⊢ Rel (∥r‘𝑅) |
3 | reldvdsr.1 | . . 3 ⊢ ∥ = (∥r‘𝑅) | |
4 | 3 | releqi 5645 | . 2 ⊢ (Rel ∥ ↔ Rel (∥r‘𝑅)) |
5 | 2, 4 | mpbir 233 | 1 ⊢ Rel ∥ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1531 ∈ wcel 2108 ∃wrex 3137 Vcvv 3493 Rel wrel 5553 ‘cfv 6348 (class class class)co 7148 Basecbs 16475 .rcmulr 16558 ∥rcdsr 19380 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fv 6356 df-dvdsr 19383 |
This theorem is referenced by: dvdsr 19388 isunit 19399 subrgdvds 19541 |
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