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Theorem eqvrelim 36714
Description: Equivalence relation implies that the domain and the range are equal. (Contributed by Peter Mazsa, 29-Dec-2021.)
Assertion
Ref Expression
eqvrelim ( EqvRel 𝑅 → dom 𝑅 = ran 𝑅)

Proof of Theorem eqvrelim
StepHypRef Expression
1 eqvrelsymrel 36712 . 2 ( EqvRel 𝑅 → SymRel 𝑅)
2 symrelim 36673 . 2 ( SymRel 𝑅 → dom 𝑅 = ran 𝑅)
31, 2syl 17 1 ( EqvRel 𝑅 → dom 𝑅 = ran 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  dom cdm 5589  ran crn 5590   SymRel wsymrel 36345   EqvRel weqvrel 36350
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601  df-symrel 36658  df-eqvrel 36698
This theorem is referenced by:  erim2  36789
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