Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cnvresrn Structured version   Visualization version   GIF version

Theorem cnvresrn 37673
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
StepHypRef Expression
1 df-rn 5677 . . 3 ran 𝑅 = dom 𝑅
21reseq2i 5968 . 2 (𝑅 ↾ ran 𝑅) = (𝑅 ↾ dom 𝑅)
3 relcnv 6093 . . 3 Rel 𝑅
4 dfrel5 37671 . . 3 (Rel 𝑅 ↔ (𝑅 ↾ dom 𝑅) = 𝑅)
53, 4mpbi 229 . 2 (𝑅 ↾ dom 𝑅) = 𝑅
62, 5eqtri 2752 1 (𝑅 ↾ ran 𝑅) = 𝑅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  ccnv 5665  dom cdm 5666  ran crn 5667  cres 5668  Rel wrel 5671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-opab 5201  df-xp 5672  df-rel 5673  df-cnv 5674  df-dm 5676  df-rn 5677  df-res 5678
This theorem is referenced by:  alrmomorn  37683
  Copyright terms: Public domain W3C validator