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Theorem coepr 31980
Description: Composition with the converse of epsilon. (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 3354 . . . . . 6 𝑥 ∈ V
31, 2brcnv 5443 . . . . 5 (𝐴 E 𝑥𝑥 E 𝐴)
41epelc 5164 . . . . 5 (𝑥 E 𝐴𝑥𝐴)
53, 4bitri 264 . . . 4 (𝐴 E 𝑥𝑥𝐴)
65anbi1i 610 . . 3 ((𝐴 E 𝑥𝑥𝑅𝐵) ↔ (𝑥𝐴𝑥𝑅𝐵))
76exbii 1924 . 2 (∃𝑥(𝐴 E 𝑥𝑥𝑅𝐵) ↔ ∃𝑥(𝑥𝐴𝑥𝑅𝐵))
8 coep.2 . . 3 𝐵 ∈ V
91, 8brco 5431 . 2 (𝐴(𝑅 E )𝐵 ↔ ∃𝑥(𝐴 E 𝑥𝑥𝑅𝐵))
10 df-rex 3067 . 2 (∃𝑥𝐴 𝑥𝑅𝐵 ↔ ∃𝑥(𝑥𝐴𝑥𝑅𝐵))
117, 9, 103bitr4i 292 1 (𝐴(𝑅 E )𝐵 ↔ ∃𝑥𝐴 𝑥𝑅𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382  wex 1852  wcel 2145  wrex 3062  Vcvv 3351   class class class wbr 4786   E cep 5161  ccnv 5248  ccom 5253
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 4915  ax-nul 4923  ax-pr 5034
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 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-eprel 5162  df-cnv 5257  df-co 5258
This theorem is referenced by:  elfuns  32359  brub  32398
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