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

Theorem cnvepimaex 37864
Description: The image of converse epsilon exists, proof via imaexALTV 37858 (see also cnvepima 37865 and uniexg 7743 for alternate way). (Contributed by Peter Mazsa, 22-Mar-2023.)
Assertion
Ref Expression
cnvepimaex (𝐴𝑉 → ( E “ 𝐴) ∈ V)

Proof of Theorem cnvepimaex
StepHypRef Expression
1 cnvepresex 37862 . 2 (𝐴𝑉 → ( E ↾ 𝐴) ∈ V)
2 imaexALTV 37858 . . 3 (( E ∈ V ∨ (( E ↾ 𝐴) ∈ V ∧ 𝐴𝑉)) → ( E “ 𝐴) ∈ V)
32olcs 874 . 2 ((( E ↾ 𝐴) ∈ V ∧ 𝐴𝑉) → ( E “ 𝐴) ∈ V)
41, 3mpancom 686 1 (𝐴𝑉 → ( E “ 𝐴) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wcel 2098  Vcvv 3463   E cep 5575  ccnv 5671  cres 5674  cima 5675
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 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738
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 2528  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-eprel 5576  df-xp 5678  df-rel 5679  df-cnv 5680  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ec 8725  df-qs 8729
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator