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Theorem erssxp 7934
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 7920 . . 3 (𝑅 Er 𝐴 → Rel 𝑅)
2 relssdmrn 5817 . . 3 (Rel 𝑅𝑅 ⊆ (dom 𝑅 × ran 𝑅))
31, 2syl 17 . 2 (𝑅 Er 𝐴𝑅 ⊆ (dom 𝑅 × ran 𝑅))
4 erdm 7921 . . 3 (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)
5 errn 7933 . . 3 (𝑅 Er 𝐴 → ran 𝑅 = 𝐴)
64, 5xpeq12d 5297 . 2 (𝑅 Er 𝐴 → (dom 𝑅 × ran 𝑅) = (𝐴 × 𝐴))
73, 6sseqtrd 3782 1 (𝑅 Er 𝐴𝑅 ⊆ (𝐴 × 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wss 3715   × cxp 5264  dom cdm 5266  ran crn 5267  Rel wrel 5271   Er wer 7908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274  df-dm 5276  df-rn 5277  df-er 7911
This theorem is referenced by:  erex  7935  riiner  7987  efgval  18330  qtophaus  30212
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