Theorem symrelim 37429

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 37169	. 2 ⊢ ran ◡𝑅 = dom 𝑅
2		dfsymrel4 37421	. . . 4 ⊢ ( SymRel 𝑅 ↔ (◡𝑅 = 𝑅 ∧ Rel 𝑅))
3	2	simplbi 499	. . 3 ⊢ ( SymRel 𝑅 → ◡𝑅 = 𝑅)
4	3	rneqd 5938	. 2 ⊢ ( SymRel 𝑅 → ran ◡𝑅 = ran 𝑅)
5	1, 4	eqtr3id 2787	1 ⊢ ( SymRel 𝑅 → dom 𝑅 = ran 𝑅)

Syntax hints:   → wi 4   = wceq 1542  ^◡ccnv 5676  dom cdm 5677  ran crn 5678  Rel wrel 5682   SymRel wsymrel 37055

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-symrel 37414

This theorem is referenced by:  eqvrelim  37471