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Theorem coep 31979
Description: Composition with epsilon. (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 . . . . . 6 𝐵 ∈ V
21epelc 5165 . . . . 5 (𝑥 E 𝐵𝑥𝐵)
32anbi2i 609 . . . 4 ((𝐴𝑅𝑥𝑥 E 𝐵) ↔ (𝐴𝑅𝑥𝑥𝐵))
4 ancom 448 . . . 4 ((𝐴𝑅𝑥𝑥𝐵) ↔ (𝑥𝐵𝐴𝑅𝑥))
53, 4bitri 264 . . 3 ((𝐴𝑅𝑥𝑥 E 𝐵) ↔ (𝑥𝐵𝐴𝑅𝑥))
65exbii 1924 . 2 (∃𝑥(𝐴𝑅𝑥𝑥 E 𝐵) ↔ ∃𝑥(𝑥𝐵𝐴𝑅𝑥))
7 coep.1 . . 3 𝐴 ∈ V
87, 1brco 5430 . 2 (𝐴( E ∘ 𝑅)𝐵 ↔ ∃𝑥(𝐴𝑅𝑥𝑥 E 𝐵))
9 df-rex 3067 . 2 (∃𝑥𝐵 𝐴𝑅𝑥 ↔ ∃𝑥(𝑥𝐵𝐴𝑅𝑥))
106, 8, 93bitr4i 292 1 (𝐴( E ∘ 𝑅)𝐵 ↔ ∃𝑥𝐵 𝐴𝑅𝑥)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382  wex 1852  wcel 2145  wrex 3062  Vcvv 3351   class class class wbr 4787   E cep 5162  ccom 5254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788  df-opab 4848  df-eprel 5163  df-co 5259
This theorem is referenced by:  dffr5  31981  brbigcup  32342  elfuns  32359  brimage  32370
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