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

Theorem coep 33698
Description: Composition with the membership relation. (Contributed by Scott Fenton, 18-Feb-2013.)
Hypotheses
Ref Expression
coep.1 𝐴 ∈ V
coep.2 𝐵 ∈ V
Assertion
Ref Expression
coep (𝐴( E ∘ 𝑅)𝐵 ↔ ∃𝑥𝐵 𝐴𝑅𝑥)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑅

Proof of Theorem coep
StepHypRef Expression
1 coep.2 . . . . 5 𝐵 ∈ V
21epeli 5496 . . . 4 (𝑥 E 𝐵𝑥𝐵)
32anbi1ci 625 . . 3 ((𝐴𝑅𝑥𝑥 E 𝐵) ↔ (𝑥𝐵𝐴𝑅𝑥))
43exbii 1853 . 2 (∃𝑥(𝐴𝑅𝑥𝑥 E 𝐵) ↔ ∃𝑥(𝑥𝐵𝐴𝑅𝑥))
5 coep.1 . . 3 𝐴 ∈ V
65, 1brco 5776 . 2 (𝐴( E ∘ 𝑅)𝐵 ↔ ∃𝑥(𝐴𝑅𝑥𝑥 E 𝐵))
7 df-rex 3071 . 2 (∃𝑥𝐵 𝐴𝑅𝑥 ↔ ∃𝑥(𝑥𝐵𝐴𝑅𝑥))
84, 6, 73bitr4i 302 1 (𝐴( E ∘ 𝑅)𝐵 ↔ ∃𝑥𝐵 𝐴𝑅𝑥)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395  wex 1785  wcel 2109  wrex 3066  Vcvv 3430   class class class wbr 5078   E cep 5493  ccom 5592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-ne 2945  df-rex 3071  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-br 5079  df-opab 5141  df-eprel 5494  df-co 5597
This theorem is referenced by:  dffr5  33700  brbigcup  34179  elfuns  34196  brimage  34207
  Copyright terms: Public domain W3C validator