Theorem cnvresrn 35087

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 5446	. . 3 ⊢ ran 𝑅 = dom ◡𝑅
2	1	reseq2i 5723	. 2 ⊢ (◡𝑅 ↾ ran 𝑅) = (◡𝑅 ↾ dom ◡𝑅)
3		relcnv 5835	. . 3 ⊢ Rel ◡𝑅
4		dfrel5 35085	. . 3 ⊢ (Rel ◡𝑅 ↔ (◡𝑅 ↾ dom ◡𝑅) = ◡𝑅)
5	3, 4	mpbi 231	. 2 ⊢ (◡𝑅 ↾ dom ◡𝑅) = ◡𝑅
6	2, 5	eqtri 2817	1 ⊢ (◡𝑅 ↾ ran 𝑅) = ◡𝑅

Syntax hints:   = wceq 1520  ^◡ccnv 5434  dom cdm 5435  ran crn 5436   ↾ cres 5437  Rel wrel 5440

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-sep 5088  ax-nul 5095  ax-pr 5214

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3434  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-nul 4207  df-if 4376  df-sn 4467  df-pr 4469  df-op 4473  df-br 4957  df-opab 5019  df-xp 5441  df-rel 5442  df-cnv 5443  df-dm 5445  df-rn 5446  df-res 5447

This theorem is referenced by:  alrmomorn  35094