Theorem cnvresrn 39471

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 5666	. . 3 ⊢ ran 𝑅 = dom ◡𝑅
2	1	reseq2i 5977	. 2 ⊢ (◡𝑅 ↾ ran 𝑅) = (◡𝑅 ↾ dom ◡𝑅)
3		relcnv 6111	. . 3 ⊢ Rel ◡𝑅
4		dfrel5 39469	. . 3 ⊢ (Rel ◡𝑅 ↔ (◡𝑅 ↾ dom ◡𝑅) = ◡𝑅)
5	3, 4	mpbi 233	. 2 ⊢ (◡𝑅 ↾ dom ◡𝑅) = ◡𝑅
6	2, 5	eqtri 2786	1 ⊢ (◡𝑅 ↾ ran 𝑅) = ◡𝑅

This proof depends on syntax axioms:   = wceq 1570  ^◡ccnv 5654  dom cdm 5655  ran crn 5656   ↾ cres 5657  Rel wrel 5660

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-xp 5661  df-rel 5662  df-cnv 5663  df-dm 5665  df-rn 5666  df-res 5667

This theorem is used by:  alrmomorn  39481