Theorem relcnvexb 7873

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 7871	. 2 ⊢ (𝑅 ∈ V → ◡𝑅 ∈ V)
2		dfrel2 6147	. . 3 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅)
3		cnvexg 7871	. . . 4 ⊢ (◡𝑅 ∈ V → ◡◡𝑅 ∈ V)
4		eleq1 2828	. . . 4 ⊢ (◡◡𝑅 = 𝑅 → (◡◡𝑅 ∈ V ↔ 𝑅 ∈ V))
5	3, 4	imbitrid 245	. . 3 ⊢ (◡◡𝑅 = 𝑅 → (◡𝑅 ∈ V → 𝑅 ∈ V))
6	2, 5	sylbi 218	. 2 ⊢ (Rel 𝑅 → (◡𝑅 ∈ V → 𝑅 ∈ V))
7	1, 6	impbid2 227	1 ⊢ (Rel 𝑅 → (𝑅 ∈ V ↔ ◡𝑅 ∈ V))

Syntax hints:   → wi 4   ↔ wb 207   = wceq 1547   ∈ wcel 2119  Vcvv 3432  ^◡ccnv 5624  Rel wrel 5630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-xp 5631  df-rel 5632  df-cnv 5633  df-dm 5635  df-rn 5636

This theorem is referenced by:  f1oexrnex  7874