Theorem symrelim 38595

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

Step	Hyp	Ref	Expression
1		rncnv 38333	. 2 ⊢ ran ◡𝑅 = dom 𝑅
2		dfsymrel4 38587	. . . 4 ⊢ ( SymRel 𝑅 ↔ (◡𝑅 = 𝑅 ∧ Rel 𝑅))
3	2	simplbi 497	. . 3 ⊢ ( SymRel 𝑅 → ◡𝑅 = 𝑅)
4	3	rneqd 5878	. 2 ⊢ ( SymRel 𝑅 → ran ◡𝑅 = ran 𝑅)
5	1, 4	eqtr3id 2780	1 ⊢ ( SymRel 𝑅 → dom 𝑅 = ran 𝑅)

Syntax hints:   → wi 4   = wceq 1541  ^◡ccnv 5615  dom cdm 5616  ran crn 5617  Rel wrel 5621   SymRel wsymrel 38226

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-xp 5622  df-rel 5623  df-cnv 5624  df-dm 5626  df-rn 5627  df-res 5628  df-symrel 38580

This theorem is referenced by:  eqvrelim  38637