Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  coepr Structured version   Visualization version   GIF version

Theorem coepr 33062
Description: Composition with the converse membership relation. (Contributed by Scott Fenton, 18-Feb-2013.)
Hypotheses
Ref Expression
coep.1 𝐴 ∈ V
coep.2 𝐵 ∈ V
Assertion
Ref Expression
coepr (𝐴(𝑅 E )𝐵 ↔ ∃𝑥𝐴 𝑥𝑅𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑅

Proof of Theorem coepr
StepHypRef Expression
1 coep.1 . . . . . 6 𝐴 ∈ V
2 vex 3472 . . . . . 6 𝑥 ∈ V
31, 2brcnv 5730 . . . . 5 (𝐴 E 𝑥𝑥 E 𝐴)
41epeli 5445 . . . . 5 (𝑥 E 𝐴𝑥𝐴)
53, 4bitri 278 . . . 4 (𝐴 E 𝑥𝑥𝐴)
65anbi1i 626 . . 3 ((𝐴 E 𝑥𝑥𝑅𝐵) ↔ (𝑥𝐴𝑥𝑅𝐵))
76exbii 1849 . 2 (∃𝑥(𝐴 E 𝑥𝑥𝑅𝐵) ↔ ∃𝑥(𝑥𝐴𝑥𝑅𝐵))
8 coep.2 . . 3 𝐵 ∈ V
91, 8brco 5718 . 2 (𝐴(𝑅 E )𝐵 ↔ ∃𝑥(𝐴 E 𝑥𝑥𝑅𝐵))
10 df-rex 3136 . 2 (∃𝑥𝐴 𝑥𝑅𝐵 ↔ ∃𝑥(𝑥𝐴𝑥𝑅𝐵))
117, 9, 103bitr4i 306 1 (𝐴(𝑅 E )𝐵 ↔ ∃𝑥𝐴 𝑥𝑅𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  wex 1781  wcel 2114  wrex 3131  Vcvv 3469   class class class wbr 5042   E cep 5441  ccnv 5531  ccom 5536
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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
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 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-rex 3136  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-br 5043  df-opab 5105  df-eprel 5442  df-cnv 5540  df-co 5541
This theorem is referenced by:  elfuns  33450  brub  33489
  Copyright terms: Public domain W3C validator