Theorem dmcoss2 39742

Description: The domain of cosets is the range. (Contributed by Peter Mazsa, 27-Dec-2018.)

Assertion

Ref	Expression
dmcoss2	⊢ dom ≀ 𝑅 = ran 𝑅

Proof of Theorem dmcoss2

Step	Hyp	Ref	Expression
1		dmcoss3 39741	. 2 ⊢ dom ≀ 𝑅 = dom ◡𝑅
2		df-rn 5669	. 2 ⊢ ran 𝑅 = dom ◡𝑅
3	1, 2	eqtr4i 2789	1 ⊢ dom ≀ 𝑅 = ran 𝑅

This proof depends on syntax axioms:   = wceq 1570  ^◡ccnv 5657  dom cdm 5658  ran crn 5659   ≀ ccoss 39381

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-coss 39699

This theorem is used by:  dm1cosscnvepres  39744