Theorem cnvresrn 38947

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 5676	. . 3 ⊢ ran 𝑅 = dom ◡𝑅
2	1	reseq2i 5979	. 2 ⊢ (◡𝑅 ↾ ran 𝑅) = (◡𝑅 ↾ dom ◡𝑅)
3		relcnv 6110	. . 3 ⊢ Rel ◡𝑅
4		dfrel5 38945	. . 3 ⊢ (Rel ◡𝑅 ↔ (◡𝑅 ↾ dom ◡𝑅) = ◡𝑅)
5	3, 4	mpbi 233	. 2 ⊢ (◡𝑅 ↾ dom ◡𝑅) = ◡𝑅
6	2, 5	eqtri 2793	1 ⊢ (◡𝑅 ↾ ran 𝑅) = ◡𝑅

Syntax hints:   = wceq 1568  ^◡ccnv 5664  dom cdm 5665  ran crn 5666   ↾ cres 5667  Rel wrel 5670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-ext 2742  ax-sep 5262  ax-pr 5408

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-ral 3087  df-rex 3097  df-rab 3424  df-v 3464  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-xp 5671  df-rel 5672  df-cnv 5673  df-dm 5675  df-rn 5676  df-res 5677

This theorem is referenced by:  alrmomorn  38957