Theorem relcnvexb 7921

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

Step	Hyp	Ref	Expression
1		cnvexg 7919	. 2 ⊢ (𝑅 ∈ V → ◡𝑅 ∈ V)
2		dfrel2 6186	. . 3 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅)
3		cnvexg 7919	. . . 4 ⊢ (◡𝑅 ∈ V → ◡◡𝑅 ∈ V)
4		eleq1 2850	. . . 4 ⊢ (◡◡𝑅 = 𝑅 → (◡◡𝑅 ∈ V ↔ 𝑅 ∈ V))
5	3, 4	imbitrid 247	. . 3 ⊢ (◡◡𝑅 = 𝑅 → (◡𝑅 ∈ V → 𝑅 ∈ V))
6	2, 5	sylbi 220	. 2 ⊢ (Rel 𝑅 → (◡𝑅 ∈ V → 𝑅 ∈ V))
7	1, 6	impbid2 229	1 ⊢ (Rel 𝑅 → (𝑅 ∈ V ↔ ◡𝑅 ∈ V))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1569   ∈ wcel 2142  Vcvv 3454  ^◡ccnv 5659  Rel wrel 5665

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-xp 5666  df-rel 5667  df-cnv 5668  df-dm 5670  df-rn 5671

This theorem is used by:  f1oexrnex  7922