Theorem eqvrelim 35863

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 35861	. 2 ⊢ ( EqvRel 𝑅 → SymRel 𝑅)
2		symrelim 35822	. 2 ⊢ ( SymRel 𝑅 → dom 𝑅 = ran 𝑅)
3	1, 2	syl 17	1 ⊢ ( EqvRel 𝑅 → dom 𝑅 = ran 𝑅)

Syntax hints:   → wi 4   = wceq 1537  dom cdm 5536  ran crn 5537   SymRel wsymrel 35492   EqvRel weqvrel 35497

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5184  ax-nul 5191  ax-pr 5311

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3483  df-dif 3922  df-un 3924  df-in 3926  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-br 5048  df-opab 5110  df-xp 5542  df-rel 5543  df-cnv 5544  df-dm 5546  df-rn 5547  df-res 5548  df-symrel 35807  df-eqvrel 35847

This theorem is referenced by:  erim2  35938