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Theorem relcnvexb 7061
 Description: A relation is a set iff its converse is a set. (Contributed by FL, 3-Mar-2007.)
Assertion
Ref Expression
relcnvexb (Rel 𝑅 → (𝑅 ∈ V ↔ 𝑅 ∈ V))

Proof of Theorem relcnvexb
StepHypRef Expression
1 cnvexg 7059 . 2 (𝑅 ∈ V → 𝑅 ∈ V)
2 dfrel2 5542 . . 3 (Rel 𝑅𝑅 = 𝑅)
3 cnvexg 7059 . . . 4 (𝑅 ∈ V → 𝑅 ∈ V)
4 eleq1 2686 . . . 4 (𝑅 = 𝑅 → (𝑅 ∈ V ↔ 𝑅 ∈ V))
53, 4syl5ib 234 . . 3 (𝑅 = 𝑅 → (𝑅 ∈ V → 𝑅 ∈ V))
62, 5sylbi 207 . 2 (Rel 𝑅 → (𝑅 ∈ V → 𝑅 ∈ V))
71, 6impbid2 216 1 (Rel 𝑅 → (𝑅 ∈ V ↔ 𝑅 ∈ V))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   = wceq 1480   ∈ wcel 1987  Vcvv 3186  ◡ccnv 5073  Rel wrel 5079 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-xp 5080  df-rel 5081  df-cnv 5082  df-dm 5084  df-rn 5085 This theorem is referenced by:  f1oexrnex  7062
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