releldm - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > releldm		Structured version Visualization version GIF version

Theorem releldm 5926

Description: The first argument of a binary relation belongs to its domain. Note that 𝐴𝑅𝐵 does not imply Rel 𝑅: see for example nrelv 5777 and brv 5441. (Contributed by NM, 2-Jul-2008.)

**Assertion**
Ref	Expression
releldm	⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)

**Proof of Theorem releldm**
Step	Hyp	Ref	Expression
1		brrelex1 5704	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ V)
2		brrelex2 5705	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ V)
3		simpr 490	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴𝑅𝐵)
4		breldmg 5891	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
5	1, 2, 3, 4	syl3anc 1398	1 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∈ wcel 2145 Vcvv 3451 class class class wbr 5103 dom cdm 5651 Rel wrel 5656

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-ext 2733 ax-sep 5249 ax-pr 5391

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-sb 2100 df-clab 2740 df-cleq 2753 df-clel 2836 df-ral 3078 df-rex 3088 df-rab 3414 df-v 3453 df-dif 3900 df-un 3902 df-in 3904 df-ss 3915 df-nul 4280 df-if 4483 df-sn 4585 df-pr 4587 df-op 4591 df-br 5104 df-opab 5168 df-xp 5657 df-rel 5658 df-dm 5661

This theorem is used by: releldmb 5928 releldmi 5930 sofld 6179 funeu 6566 fnbr 6648 funbrfv2b 6943 funfvbrb 7051 ercl 8730 inviso1 17942 setciso 18267 rngciso 20891 ringciso 20925 lmle 25623 dvidlem 26236 dvmulbr 26260 dvcobr 26267 ulmcau 26723 ulmdvlem3 26730 metideq 34526 heibor1lem 38745 rrncmslem 38768 eqvrelcl 39630 ntrclsiex 45054 ntrneiiex 45077 binomcxplemnn0 45334 binomcxplemnotnn0 45341 sumnnodd 46643 climlimsup 46771 climlimsupcex 46780 climliminflimsupd 46812 liminflimsupclim 46818 dmclimxlim 46862 xlimclimdm 46865 xlimresdm 46870 ioodvbdlimc1lem2 46943 ioodvbdlimc2lem 46945 funbrafv 48229 funbrafv2b 48230 rngcisoALTV 49375 ringcisoALTV 49409 isinito3 50609

W3C validator