Theorem errn 6535

Description: The range and domain of an equivalence relation are equal. (Contributed by Rodolfo Medina, 11-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)

Assertion

Ref	Expression
errn	⊢ (𝑅 Er 𝐴 → ran 𝑅 = 𝐴)

Proof of Theorem errn

Step	Hyp	Ref	Expression
1		df-rn 4622	. 2 ⊢ ran 𝑅 = dom ◡𝑅
2		ercnv 6534	. . . 4 ⊢ (𝑅 Er 𝐴 → ◡𝑅 = 𝑅)
3	2	dmeqd 4813	. . 3 ⊢ (𝑅 Er 𝐴 → dom ◡𝑅 = dom 𝑅)
4		erdm 6523	. . 3 ⊢ (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)
5	3, 4	eqtrd 2203	. 2 ⊢ (𝑅 Er 𝐴 → dom ◡𝑅 = 𝐴)
6	1, 5	eqtrid 2215	1 ⊢ (𝑅 Er 𝐴 → ran 𝑅 = 𝐴)

Syntax hints:   → wi 4   = wceq 1348  ^◡ccnv 4610  dom cdm 4611  ran crn 4612   Er wer 6510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-cnv 4619  df-dm 4621  df-rn 4622  df-er 6513

This theorem is referenced by:  erssxp  6536  ecss  6554  uniqs2  6573