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Theorem symrelim 35913
 Description: Symmetric relation implies that the domain and the range are equal. (Contributed by Peter Mazsa, 29-Dec-2021.)
Assertion
Ref Expression
symrelim ( SymRel 𝑅 → dom 𝑅 = ran 𝑅)

Proof of Theorem symrelim
StepHypRef Expression
1 rncnv 35676 . 2 ran 𝑅 = dom 𝑅
2 dfsymrel4 35905 . . . 4 ( SymRel 𝑅 ↔ (𝑅 = 𝑅 ∧ Rel 𝑅))
32simplbi 501 . . 3 ( SymRel 𝑅𝑅 = 𝑅)
43rneqd 5785 . 2 ( SymRel 𝑅 → ran 𝑅 = ran 𝑅)
51, 4syl5eqr 2871 1 ( SymRel 𝑅 → dom 𝑅 = ran 𝑅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538  ◡ccnv 5531  dom cdm 5532  ran crn 5533  Rel wrel 5537   SymRel wsymrel 35583 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-br 5043  df-opab 5105  df-xp 5538  df-rel 5539  df-cnv 5540  df-dm 5542  df-rn 5543  df-res 5544  df-symrel 35898 This theorem is referenced by:  eqvrelim  35954
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