Theorem eqvrelim 39148

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 39146	. 2 ⊢ ( EqvRel 𝑅 → SymRel 𝑅)
2		symrelim 39106	. 2 ⊢ ( SymRel 𝑅 → dom 𝑅 = ran 𝑅)
3	1, 2	syl 17	1 ⊢ ( EqvRel 𝑅 → dom 𝑅 = ran 𝑅)

Syntax hints:   → wi 4   = wceq 1559  dom cdm 5645  ran crn 5646   SymRel wsymrel 38658   EqvRel weqvrel 38663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-xp 5651  df-rel 5652  df-cnv 5653  df-dm 5655  df-rn 5656  df-res 5657  df-symrel 39087  df-eqvrel 39132

This theorem is referenced by:  erimeq2  39226