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Mirrors > Home > MPE Home > Th. List > releldm | Structured version Visualization version GIF version |
Description: The first argument of a binary relation belongs to its domain. Note that 𝐴𝑅𝐵 does not imply Rel 𝑅: see for example nrelv 5383 and brv 5068. (Contributed by NM, 2-Jul-2008.) |
Ref | Expression |
---|---|
releldm | ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brrelex 5296 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ V) | |
2 | brrelex2 5297 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ V) | |
3 | simpr 471 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴𝑅𝐵) | |
4 | breldmg 5468 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅) | |
5 | 1, 2, 3, 4 | syl3anc 1476 | 1 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 ∈ wcel 2145 Vcvv 3351 class class class wbr 4786 dom cdm 5249 Rel wrel 5254 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pr 5034 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-br 4787 df-opab 4847 df-xp 5255 df-rel 5256 df-dm 5259 |
This theorem is referenced by: releldmb 5498 releldmi 5500 sofld 5722 funeu 6056 fnbr 6133 funbrfv2b 6382 funfvbrb 6473 ercl 7907 inviso1 16633 setciso 16948 lmle 23318 dvidlem 23899 dvmulbr 23922 dvcobr 23929 ulmcau 24369 ulmdvlem3 24376 metideq 30276 heibor1lem 33940 rrncmslem 33963 ntrclsiex 38877 ntrneiiex 38900 binomcxplemnn0 39074 binomcxplemnotnn0 39081 sumnnodd 40380 climlimsup 40510 climlimsupcex 40519 climliminflimsupd 40551 liminflimsupclim 40557 ioodvbdlimc1lem2 40665 ioodvbdlimc2lem 40667 funbrafv 41758 funbrafv2b 41759 rngciso 42510 rngcisoALTV 42522 ringciso 42561 ringcisoALTV 42585 |
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