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Theorem erssxp 8672
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 8658 . . 3 (𝑅 Er 𝐴 → Rel 𝑅)
2 relssdmrn 6221 . . 3 (Rel 𝑅𝑅 ⊆ (dom 𝑅 × ran 𝑅))
31, 2syl 17 . 2 (𝑅 Er 𝐴𝑅 ⊆ (dom 𝑅 × ran 𝑅))
4 erdm 8659 . . 3 (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)
5 errn 8671 . . 3 (𝑅 Er 𝐴 → ran 𝑅 = 𝐴)
64, 5xpeq12d 5665 . 2 (𝑅 Er 𝐴 → (dom 𝑅 × ran 𝑅) = (𝐴 × 𝐴))
73, 6sseqtrd 3985 1 (𝑅 Er 𝐴𝑅 ⊆ (𝐴 × 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wss 3911   × cxp 5632  dom cdm 5634  ran crn 5635  Rel wrel 5639   Er wer 8646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-xp 5640  df-rel 5641  df-cnv 5642  df-dm 5644  df-rn 5645  df-er 8649
This theorem is referenced by:  erex  8673  riiner  8730  efgval  19500  qtophaus  32420
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