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Theorem erssxp 8732
Description: An equivalence relation is a subset of the cartesian product of the field. (Contributed by Mario Carneiro, 12-Aug-2015.)
Assertion
Ref Expression
erssxp (𝑅 Er 𝐴𝑅 ⊆ (𝐴 × 𝐴))

Proof of Theorem erssxp
StepHypRef Expression
1 errel 8718 . . 3 (𝑅 Er 𝐴 → Rel 𝑅)
2 relssdmrn 6267 . . 3 (Rel 𝑅𝑅 ⊆ (dom 𝑅 × ran 𝑅))
31, 2syl 17 . 2 (𝑅 Er 𝐴𝑅 ⊆ (dom 𝑅 × ran 𝑅))
4 erdm 8719 . . 3 (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)
5 errn 8731 . . 3 (𝑅 Er 𝐴 → ran 𝑅 = 𝐴)
64, 5xpeq12d 5707 . 2 (𝑅 Er 𝐴 → (dom 𝑅 × ran 𝑅) = (𝐴 × 𝐴))
73, 6sseqtrd 4022 1 (𝑅 Er 𝐴𝑅 ⊆ (𝐴 × 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wss 3948   × cxp 5674  dom cdm 5676  ran crn 5677  Rel wrel 5681   Er wer 8706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684  df-dm 5686  df-rn 5687  df-er 8709
This theorem is referenced by:  erex  8733  riiner  8790  efgval  19633  qtophaus  33280
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