Theorem symrelim 38888

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 38551	. 2 ⊢ ran ◡𝑅 = dom 𝑅
2		dfsymrel4 38880	. . . 4 ⊢ ( SymRel 𝑅 ↔ (◡𝑅 = 𝑅 ∧ Rel 𝑅))
3	2	simplbi 496	. . 3 ⊢ ( SymRel 𝑅 → ◡𝑅 = 𝑅)
4	3	rneqd 5895	. 2 ⊢ ( SymRel 𝑅 → ran ◡𝑅 = ran 𝑅)
5	1, 4	eqtr3id 2786	1 ⊢ ( SymRel 𝑅 → dom 𝑅 = ran 𝑅)

Syntax hints:   → wi 4   = wceq 1542  ^◡ccnv 5631  dom cdm 5632  ran crn 5633  Rel wrel 5637   SymRel wsymrel 38440

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5638  df-rel 5639  df-cnv 5640  df-dm 5642  df-rn 5643  df-res 5644  df-symrel 38869

This theorem is referenced by:  eqvrelim  38930