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Theorem erssxp 6315
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 6301 . . 3 (𝑅 Er 𝐴 → Rel 𝑅)
2 relssdmrn 4951 . . 3 (Rel 𝑅𝑅 ⊆ (dom 𝑅 × ran 𝑅))
31, 2syl 14 . 2 (𝑅 Er 𝐴𝑅 ⊆ (dom 𝑅 × ran 𝑅))
4 erdm 6302 . . 3 (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)
5 errn 6314 . . 3 (𝑅 Er 𝐴 → ran 𝑅 = 𝐴)
64, 5xpeq12d 4463 . 2 (𝑅 Er 𝐴 → (dom 𝑅 × ran 𝑅) = (𝐴 × 𝐴))
73, 6sseqtrd 3062 1 (𝑅 Er 𝐴𝑅 ⊆ (𝐴 × 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wss 2999   × cxp 4436  dom cdm 4438  ran crn 4439  Rel wrel 4443   Er wer 6289
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-xp 4444  df-rel 4445  df-cnv 4446  df-dm 4448  df-rn 4449  df-er 6292
This theorem is referenced by:  erex  6316  riinerm  6365
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