Theorem errn 8313

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 5568	. 2 ⊢ ran 𝑅 = dom ◡𝑅
2		ercnv 8312	. . . 4 ⊢ (𝑅 Er 𝐴 → ◡𝑅 = 𝑅)
3	2	dmeqd 5776	. . 3 ⊢ (𝑅 Er 𝐴 → dom ◡𝑅 = dom 𝑅)
4		erdm 8301	. . 3 ⊢ (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)
5	3, 4	eqtrd 2858	. 2 ⊢ (𝑅 Er 𝐴 → dom ◡𝑅 = 𝐴)
6	1, 5	syl5eq 2870	1 ⊢ (𝑅 Er 𝐴 → ran 𝑅 = 𝐴)

Syntax hints:   → wi 4   = wceq 1537  ^◡ccnv 5556  dom cdm 5557  ran crn 5558   Er wer 8288

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 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

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 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-br 5069  df-opab 5131  df-xp 5563  df-rel 5564  df-cnv 5565  df-dm 5567  df-rn 5568  df-er 8291

This theorem is referenced by:  erssxp  8314  ecss  8337  uniqs2  8361  sylow2a  18746