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Theorem relcnvexb 7624
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 7622 . 2 (𝑅 ∈ V → 𝑅 ∈ V)
2 dfrel2 6045 . . 3 (Rel 𝑅𝑅 = 𝑅)
3 cnvexg 7622 . . . 4 (𝑅 ∈ V → 𝑅 ∈ V)
4 eleq1 2905 . . . 4 (𝑅 = 𝑅 → (𝑅 ∈ V ↔ 𝑅 ∈ V))
53, 4syl5ib 245 . . 3 (𝑅 = 𝑅 → (𝑅 ∈ V → 𝑅 ∈ V))
62, 5sylbi 218 . 2 (Rel 𝑅 → (𝑅 ∈ V → 𝑅 ∈ V))
71, 6impbid2 227 1 (Rel 𝑅 → (𝑅 ∈ V ↔ 𝑅 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1530  wcel 2107  Vcvv 3500  ccnv 5553  Rel wrel 5559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-xp 5560  df-rel 5561  df-cnv 5562  df-dm 5564  df-rn 5565
This theorem is referenced by:  f1oexrnex  7625
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