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Theorem erssxp 6217
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 6203 . . 3 (𝑅 Er 𝐴 → Rel 𝑅)
2 relssdmrn 4892 . . 3 (Rel 𝑅𝑅 ⊆ (dom 𝑅 × ran 𝑅))
31, 2syl 14 . 2 (𝑅 Er 𝐴𝑅 ⊆ (dom 𝑅 × ran 𝑅))
4 erdm 6204 . . 3 (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)
5 errn 6216 . . 3 (𝑅 Er 𝐴 → ran 𝑅 = 𝐴)
64, 5xpeq12d 4417 . 2 (𝑅 Er 𝐴 → (dom 𝑅 × ran 𝑅) = (𝐴 × 𝐴))
73, 6sseqtrd 3045 1 (𝑅 Er 𝐴𝑅 ⊆ (𝐴 × 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wss 2983   × cxp 4390  dom cdm 4392  ran crn 4393  Rel wrel 4397   Er wer 6191
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917  ax-pow 3969  ax-pr 3993
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-br 3807  df-opab 3861  df-xp 4398  df-rel 4399  df-cnv 4400  df-dm 4402  df-rn 4403  df-er 6194
This theorem is referenced by:  erex  6218  riinerm  6267
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