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Theorem relssdmrn 6266
Description: A relation is included in the Cartesian product of its domain and range. Exercise 4.12(t) of [Mendelson] p. 235. (Contributed by NM, 3-Aug-1994.) (Proof shortened by SN, 23-Dec-2024.)
Assertion
Ref Expression
relssdmrn (Rel 𝐴𝐴 ⊆ (dom 𝐴 × ran 𝐴))

Proof of Theorem relssdmrn
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 22 . 2 (Rel 𝐴 → Rel 𝐴)
2 vex 3473 . . . . 5 𝑥 ∈ V
3 vex 3473 . . . . 5 𝑦 ∈ V
42, 3opeldm 5904 . . . 4 (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥 ∈ dom 𝐴)
52, 3opelrn 5939 . . . 4 (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 ∈ ran 𝐴)
64, 5opelxpd 5711 . . 3 (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ (dom 𝐴 × ran 𝐴))
76a1i 11 . 2 (Rel 𝐴 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ (dom 𝐴 × ran 𝐴)))
81, 7relssdv 5784 1 (Rel 𝐴𝐴 ⊆ (dom 𝐴 × ran 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2099  wss 3944  cop 4630   × cxp 5670  dom cdm 5672  ran crn 5673  Rel wrel 5677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-xp 5678  df-rel 5679  df-cnv 5680  df-dm 5682  df-rn 5683
This theorem is referenced by:  resssxp  6268  cnvssrndm  6269  cossxp  6270  relrelss  6271  relfld  6273  fssxp  6745  oprabss  7521  cnvexg  7926  resfunexgALT  7945  cofunexg  7946  fnexALT  7948  funexw  7949  erssxp  8741  ttrclexg  9738  wunco  10748  trclublem  14966  trclubi  14967  trclub  14969  reltrclfv  14988  imasless  17513  sylow2a  19565  gsum2d  19918  znleval  21475  tsmsxp  24046  relfi  32377  fcnvgreu  32442  relssinxpdmrn  37757  trclubNEW  42972  trrelsuperreldg  43021  trrelsuperrel2dg  43024
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