Theorem relcnvexb 5276

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 5274	. 2 ⊢ (𝑅 ∈ V → ◡𝑅 ∈ V)
2		dfrel2 5187	. . 3 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅)
3		cnvexg 5274	. . . 4 ⊢ (◡𝑅 ∈ V → ◡◡𝑅 ∈ V)
4		eleq1 2294	. . . 4 ⊢ (◡◡𝑅 = 𝑅 → (◡◡𝑅 ∈ V ↔ 𝑅 ∈ V))
5	3, 4	imbitrid 154	. . 3 ⊢ (◡◡𝑅 = 𝑅 → (◡𝑅 ∈ V → 𝑅 ∈ V))
6	2, 5	sylbi 121	. 2 ⊢ (Rel 𝑅 → (◡𝑅 ∈ V → 𝑅 ∈ V))
7	1, 6	impbid2 143	1 ⊢ (Rel 𝑅 → (𝑅 ∈ V ↔ ◡𝑅 ∈ V))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1397   ∈ wcel 2202  Vcvv 2802  ^◡ccnv 4724  Rel wrel 4730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-xp 4731  df-rel 4732  df-cnv 4733  df-dm 4735  df-rn 4736

This theorem is referenced by: (None)