Theorem releldm 5496

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.)

Assertion

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

Proof of Theorem releldm

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)

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