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Theorem cnvresrn 35758
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 5534 . . 3 ran 𝑅 = dom 𝑅
21reseq2i 5819 . 2 (𝑅 ↾ ran 𝑅) = (𝑅 ↾ dom 𝑅)
3 relcnv 5938 . . 3 Rel 𝑅
4 dfrel5 35756 . . 3 (Rel 𝑅 ↔ (𝑅 ↾ dom 𝑅) = 𝑅)
53, 4mpbi 233 . 2 (𝑅 ↾ dom 𝑅) = 𝑅
62, 5eqtri 2824 1 (𝑅 ↾ ran 𝑅) = 𝑅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  ccnv 5522  dom cdm 5523  ran crn 5524  cres 5525  Rel wrel 5528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-xp 5529  df-rel 5530  df-cnv 5531  df-dm 5533  df-rn 5534  df-res 5535
This theorem is referenced by:  alrmomorn  35765
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