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Mirrors > Home > ILE Home > Th. List > relcnvexb | GIF version |
Description: A relation is a set iff its converse is a set. (Contributed by FL, 3-Mar-2007.) |
Ref | Expression |
---|---|
relcnvexb | ⊢ (Rel 𝑅 → (𝑅 ∈ V ↔ ◡𝑅 ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvexg 5122 | . 2 ⊢ (𝑅 ∈ V → ◡𝑅 ∈ V) | |
2 | dfrel2 5035 | . . 3 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅) | |
3 | cnvexg 5122 | . . . 4 ⊢ (◡𝑅 ∈ V → ◡◡𝑅 ∈ V) | |
4 | eleq1 2220 | . . . 4 ⊢ (◡◡𝑅 = 𝑅 → (◡◡𝑅 ∈ V ↔ 𝑅 ∈ V)) | |
5 | 3, 4 | syl5ib 153 | . . 3 ⊢ (◡◡𝑅 = 𝑅 → (◡𝑅 ∈ V → 𝑅 ∈ V)) |
6 | 2, 5 | sylbi 120 | . 2 ⊢ (Rel 𝑅 → (◡𝑅 ∈ V → 𝑅 ∈ V)) |
7 | 1, 6 | impbid2 142 | 1 ⊢ (Rel 𝑅 → (𝑅 ∈ V ↔ ◡𝑅 ∈ V)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1335 ∈ wcel 2128 Vcvv 2712 ◡ccnv 4584 Rel wrel 4590 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-opab 4026 df-xp 4591 df-rel 4592 df-cnv 4593 df-dm 4595 df-rn 4596 |
This theorem is referenced by: (None) |
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