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Theorem symrelim 37942
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 37682 . 2 ran 𝑅 = dom 𝑅
2 dfsymrel4 37934 . . . 4 ( SymRel 𝑅 ↔ (𝑅 = 𝑅 ∧ Rel 𝑅))
32simplbi 497 . . 3 ( SymRel 𝑅𝑅 = 𝑅)
43rneqd 5931 . 2 ( SymRel 𝑅 → ran 𝑅 = ran 𝑅)
51, 4eqtr3id 2780 1 ( SymRel 𝑅 → dom 𝑅 = ran 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  ccnv 5668  dom cdm 5669  ran crn 5670  Rel wrel 5674   SymRel wsymrel 37568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-symrel 37927
This theorem is referenced by:  eqvrelim  37984
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