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Theorem relcnvexb 5124
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 5122 . 2 (𝑅 ∈ V → 𝑅 ∈ V)
2 dfrel2 5035 . . 3 (Rel 𝑅𝑅 = 𝑅)
3 cnvexg 5122 . . . 4 (𝑅 ∈ V → 𝑅 ∈ V)
4 eleq1 2220 . . . 4 (𝑅 = 𝑅 → (𝑅 ∈ V ↔ 𝑅 ∈ V))
53, 4syl5ib 153 . . 3 (𝑅 = 𝑅 → (𝑅 ∈ V → 𝑅 ∈ V))
62, 5sylbi 120 . 2 (Rel 𝑅 → (𝑅 ∈ V → 𝑅 ∈ V))
71, 6impbid2 142 1 (Rel 𝑅 → (𝑅 ∈ V ↔ 𝑅 ∈ V))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1335  wcel 2128  Vcvv 2712  ccnv 4584  Rel wrel 4590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-xp 4591  df-rel 4592  df-cnv 4593  df-dm 4595  df-rn 4596
This theorem is referenced by: (None)
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