Theorem cnvresrn 36483

Description: Converse restricted to range is converse. (Contributed by Peter Mazsa, 3-Sep-2021.)

Assertion

Ref	Expression
cnvresrn	⊢ (◡𝑅 ↾ ran 𝑅) = ◡𝑅

Proof of Theorem cnvresrn

Step	Hyp	Ref	Expression
1		df-rn 5600	. . 3 ⊢ ran 𝑅 = dom ◡𝑅
2	1	reseq2i 5888	. 2 ⊢ (◡𝑅 ↾ ran 𝑅) = (◡𝑅 ↾ dom ◡𝑅)
3		relcnv 6012	. . 3 ⊢ Rel ◡𝑅
4		dfrel5 36481	. . 3 ⊢ (Rel ◡𝑅 ↔ (◡𝑅 ↾ dom ◡𝑅) = ◡𝑅)
5	3, 4	mpbi 229	. 2 ⊢ (◡𝑅 ↾ dom ◡𝑅) = ◡𝑅
6	2, 5	eqtri 2766	1 ⊢ (◡𝑅 ↾ ran 𝑅) = ◡𝑅

Syntax hints:   = wceq 1539  ^◡ccnv 5588  dom cdm 5589  ran crn 5590   ↾ cres 5591  Rel wrel 5594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601

This theorem is referenced by:  alrmomorn  36490