Theorem dmcoss2 38879

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 38878	. 2 ⊢ dom ≀ 𝑅 = dom ◡𝑅
2		df-rn 5635	. 2 ⊢ ran 𝑅 = dom ◡𝑅
3	1, 2	eqtr4i 2763	1 ⊢ dom ≀ 𝑅 = ran 𝑅

Syntax hints:   = wceq 1542  ^◡ccnv 5623  dom cdm 5624  ran crn 5625   ≀ ccoss 38518

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-coss 38836

This theorem is referenced by:  dm1cosscnvepres  38881