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

Theorem cnvepima 37841
Description: The image of converse epsilon. (Contributed by Peter Mazsa, 22-Mar-2023.)
Assertion
Ref Expression
cnvepima (𝐴𝑉 → ( E “ 𝐴) = 𝐴)

Proof of Theorem cnvepima
StepHypRef Expression
1 cnvepresex 37838 . . 3 (𝐴𝑉 → ( E ↾ 𝐴) ∈ V)
2 uniqsALTV 37833 . . 3 (( E ↾ 𝐴) ∈ V → (𝐴 / E ) = ( E “ 𝐴))
31, 2syl 17 . 2 (𝐴𝑉 (𝐴 / E ) = ( E “ 𝐴))
4 qsid 8808 . . 3 (𝐴 / E ) = 𝐴
54unieqi 4924 . 2 (𝐴 / E ) = 𝐴
63, 5eqtr3di 2783 1 (𝐴𝑉 → ( E “ 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  Vcvv 3473   cuni 4912   E cep 5585  ccnv 5681  cres 5684  cima 5685   / cqs 8730
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-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-eprel 5586  df-xp 5688  df-rel 5689  df-cnv 5690  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ec 8733  df-qs 8737
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator