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| Mirrors > Home > MPE Home > Th. List > releldmi | Structured version Visualization version GIF version | ||
| Description: The first argument of a binary relation belongs to its domain. (Contributed by NM, 28-Apr-2015.) |
| Ref | Expression |
|---|---|
| releldm.1 | ⊢ Rel 𝑅 |
| Ref | Expression |
|---|---|
| releldmi | ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ dom 𝑅) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | releldm.1 | . 2 ⊢ Rel 𝑅 | |
| 2 | releldm 5955 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅) | |
| 3 | 1, 2 | mpan 690 | 1 ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ dom 𝑅) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 class class class wbr 5143 dom cdm 5685 Rel wrel 5690 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-dm 5695 |
| This theorem is referenced by: fpwwe2lem10 10680 fpwwe2lem11 10681 fpwwe2lem12 10682 rlimpm 15536 rlimdm 15587 iserex 15693 caucvgrlem2 15711 caucvgr 15712 caurcvg2 15714 caucvg 15715 fsumcvg3 15765 cvgcmpce 15854 climcnds 15887 trirecip 15899 ledm 18635 cmetcaulem 25322 ovoliunlem1 25537 mbflimlem 25702 dvaddf 25979 dvmulf 25980 dvcof 25986 dvcnv 26015 abelthlem5 26479 emcllem6 27044 lgamgulmlem4 27075 hlimcaui 31255 brfvrcld2 43705 sumnnodd 45645 climliminf 45821 stirlinglem12 46100 fouriersw 46246 rlimdmafv 47189 rlimdmafv2 47270 |
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