![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > relelrni | Structured version Visualization version GIF version |
Description: The second argument of a binary relation belongs to its range. (Contributed by NM, 28-Apr-2015.) |
Ref | Expression |
---|---|
releldm.1 | ⊢ Rel 𝑅 |
Ref | Expression |
---|---|
relelrni | ⊢ (𝐴𝑅𝐵 → 𝐵 ∈ ran 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | releldm.1 | . 2 ⊢ Rel 𝑅 | |
2 | relelrn 5593 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ ran 𝑅) | |
3 | 1, 2 | mpan 683 | 1 ⊢ (𝐴𝑅𝐵 → 𝐵 ∈ ran 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2166 class class class wbr 4874 ran crn 5344 Rel wrel 5348 |
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-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pr 5128 |
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 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ral 3123 df-rex 3124 df-rab 3127 df-v 3417 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-br 4875 df-opab 4937 df-xp 5349 df-rel 5350 df-cnv 5351 df-dm 5353 df-rn 5354 |
This theorem is referenced by: fpwwe2lem12 9779 lern 17579 brres2 34587 brfvrcld2 38826 |
Copyright terms: Public domain | W3C validator |