Theorem cnvresrn 39025

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 5671	. . 3 ⊢ ran 𝑅 = dom ◡𝑅
2	1	reseq2i 5974	. 2 ⊢ (◡𝑅 ↾ ran 𝑅) = (◡𝑅 ↾ dom ◡𝑅)
3		relcnv 6105	. . 3 ⊢ Rel ◡𝑅
4		dfrel5 39023	. . 3 ⊢ (Rel ◡𝑅 ↔ (◡𝑅 ↾ dom ◡𝑅) = ◡𝑅)
5	3, 4	mpbi 233	. 2 ⊢ (◡𝑅 ↾ dom ◡𝑅) = ◡𝑅
6	2, 5	eqtri 2785	1 ⊢ (◡𝑅 ↾ ran 𝑅) = ◡𝑅

This proof depends on syntax axioms:   = wceq 1569  ^◡ccnv 5659  dom cdm 5660  ran crn 5661   ↾ cres 5662  Rel wrel 5665

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-xp 5666  df-rel 5667  df-cnv 5668  df-dm 5670  df-rn 5671  df-res 5672

This theorem is used by:  alrmomorn  39035