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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 5279 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅) | |
3 | 1, 2 | mpan 702 | 1 ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ dom 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 class class class wbr 4583 dom cdm 5038 Rel wrel 5043 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-dm 5048 |
This theorem is referenced by: fpwwe2lem11 9341 fpwwe2lem12 9342 fpwwe2lem13 9343 rlimpm 14079 rlimdm 14130 iserex 14235 caucvgrlem2 14253 caucvgr 14254 caurcvg2 14256 caucvg 14257 fsumcvg3 14307 cvgcmpce 14391 climcnds 14422 trirecip 14434 ledm 17047 cmetcaulem 22894 ovoliunlem1 23077 mbflimlem 23240 dvaddf 23511 dvmulf 23512 dvcof 23517 dvcnv 23544 abelthlem5 23993 emcllem6 24527 lgamgulmlem4 24558 hlimcaui 27477 brfvrcld2 37003 sumnnodd 38697 stirlinglem12 38978 fouriersw 39124 rlimdmafv 39906 |
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