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Theorem relcnvexb 7261
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 7259 . 2 (𝑅 ∈ V → 𝑅 ∈ V)
2 dfrel2 5724 . . 3 (Rel 𝑅𝑅 = 𝑅)
3 cnvexg 7259 . . . 4 (𝑅 ∈ V → 𝑅 ∈ V)
4 eleq1 2838 . . . 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 1631  wcel 2145  Vcvv 3351  ccnv 5248  Rel wrel 5254
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-8 2147  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-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  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-ral 3066  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-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-xp 5255  df-rel 5256  df-cnv 5257  df-dm 5259  df-rn 5260
This theorem is referenced by:  f1oexrnex  7262
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