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Theorem relcnvexb 4857
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 4855 . 2 (𝑅 ∈ V → 𝑅 ∈ V)
2 dfrel2 4771 . . 3 (Rel 𝑅𝑅 = 𝑅)
3 cnvexg 4855 . . . 4 (𝑅 ∈ V → 𝑅 ∈ V)
4 eleq1 2100 . . . 4 (𝑅 = 𝑅 → (𝑅 ∈ V ↔ 𝑅 ∈ V))
53, 4syl5ib 143 . . 3 (𝑅 = 𝑅 → (𝑅 ∈ V → 𝑅 ∈ V))
62, 5sylbi 114 . 2 (Rel 𝑅 → (𝑅 ∈ V → 𝑅 ∈ V))
71, 6impbid2 131 1 (Rel 𝑅 → (𝑅 ∈ V ↔ 𝑅 ∈ V))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1243  wcel 1393  Vcvv 2557  ccnv 4344  Rel wrel 4350
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-dm 4355  df-rn 4356
This theorem is referenced by: (None)
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