Theorem eqvrelim 39585

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 39583	. 2 ⊢ ( EqvRel 𝑅 → SymRel 𝑅)
2		symrelim 39543	. 2 ⊢ ( SymRel 𝑅 → dom 𝑅 = ran 𝑅)
3	1, 2	syl 18	1 ⊢ ( EqvRel 𝑅 → dom 𝑅 = ran 𝑅)

This proof depends on syntax axioms:   → wi 4   = wceq 1570  dom cdm 5651  ran crn 5652   SymRel wsymrel 39095   EqvRel weqvrel 39100

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5657  df-rel 5658  df-cnv 5659  df-dm 5661  df-rn 5662  df-res 5663  df-symrel 39524  df-eqvrel 39569

This theorem is used by:  erimeq2  39663