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Theorem coep 35218
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 5573 . . . 4 (𝑥 E 𝐵𝑥𝐵)
32anbi1ci 625 . . 3 ((𝐴𝑅𝑥𝑥 E 𝐵) ↔ (𝑥𝐵𝐴𝑅𝑥))
43exbii 1842 . 2 (∃𝑥(𝐴𝑅𝑥𝑥 E 𝐵) ↔ ∃𝑥(𝑥𝐵𝐴𝑅𝑥))
5 coep.1 . . 3 𝐴 ∈ V
65, 1brco 5861 . 2 (𝐴( E ∘ 𝑅)𝐵 ↔ ∃𝑥(𝐴𝑅𝑥𝑥 E 𝐵))
7 df-rex 3063 . 2 (∃𝑥𝐵 𝐴𝑅𝑥 ↔ ∃𝑥(𝑥𝐵𝐴𝑅𝑥))
84, 6, 73bitr4i 303 1 (𝐴( E ∘ 𝑅)𝐵 ↔ ∃𝑥𝐵 𝐴𝑅𝑥)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395  wex 1773  wcel 2098  wrex 3062  Vcvv 3466   class class class wbr 5139   E cep 5570  ccom 5671
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-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-eprel 5571  df-co 5676
This theorem is referenced by:  dffr5  35220  brbigcup  35366  elfuns  35383  brimage  35394
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