Mathbox for Richard Penner < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  imanonrel Structured version   Visualization version   GIF version

Theorem imanonrel 37419
 Description: An image under the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.)
Assertion
Ref Expression
imanonrel ((𝐴𝐴) “ 𝐵) = ∅

Proof of Theorem imanonrel
StepHypRef Expression
1 df-ima 5097 . 2 ((𝐴𝐴) “ 𝐵) = ran ((𝐴𝐴) ↾ 𝐵)
2 resnonrel 37418 . . 3 ((𝐴𝐴) ↾ 𝐵) = ∅
32rneqi 5322 . 2 ran ((𝐴𝐴) ↾ 𝐵) = ran ∅
4 rn0 5347 . 2 ran ∅ = ∅
51, 3, 43eqtri 2647 1 ((𝐴𝐴) “ 𝐵) = ∅
 Colors of variables: wff setvar class Syntax hints:   = wceq 1480   ∖ cdif 3557  ∅c0 3897  ◡ccnv 5083  ran crn 5085   ↾ cres 5086   “ cima 5087 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624  df-opab 4684  df-xp 5090  df-rel 5091  df-cnv 5092  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator