Theorem eqvrelim 36620

Description: Equivalence relation implies that the domain and the range are equal. (Contributed by Peter Mazsa, 29-Dec-2021.)

Assertion

Ref	Expression
eqvrelim	⊢ ( EqvRel 𝑅 → dom 𝑅 = ran 𝑅)

Proof of Theorem eqvrelim

Step	Hyp	Ref	Expression
1		eqvrelsymrel 36618	. 2 ⊢ ( EqvRel 𝑅 → SymRel 𝑅)
2		symrelim 36579	. 2 ⊢ ( SymRel 𝑅 → dom 𝑅 = ran 𝑅)
3	1, 2	syl 17	1 ⊢ ( EqvRel 𝑅 → dom 𝑅 = ran 𝑅)

Syntax hints:   → wi 4   = wceq 1543  dom cdm 5579  ran crn 5580   SymRel wsymrel 36251   EqvRel weqvrel 36256

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2710  ax-sep 5216  ax-nul 5223  ax-pr 5346

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-clab 2717  df-cleq 2731  df-clel 2818  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3425  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4255  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5585  df-rel 5586  df-cnv 5587  df-dm 5589  df-rn 5590  df-res 5591  df-symrel 36564  df-eqvrel 36604

This theorem is referenced by:  erim2  36695