Theorem symrelim 39668

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 39331	. 2 ⊢ ran ◡𝑅 = dom 𝑅
2		dfsymrel4 39660	. . . 4 ⊢ ( SymRel 𝑅 ↔ (◡𝑅 = 𝑅 ∧ Rel 𝑅))
3	2	simplbi 502	. . 3 ⊢ ( SymRel 𝑅 → ◡𝑅 = 𝑅)
4	3	rneqd 5919	. 2 ⊢ ( SymRel 𝑅 → ran ◡𝑅 = ran 𝑅)
5	1, 4	eqtr3id 2809	1 ⊢ ( SymRel 𝑅 → dom 𝑅 = ran 𝑅)

This proof depends on syntax axioms:   → wi 4   = wceq 1569  ^◡ccnv 5649  dom cdm 5650  ran crn 5651  Rel wrel 5655   SymRel wsymrel 39220

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-xp 5656  df-rel 5657  df-cnv 5658  df-dm 5660  df-rn 5661  df-res 5662  df-symrel 39649

This theorem is used by:  eqvrelim  39710