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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 4875 | . 2 ⊢ (𝑅 ∈ V → ◡𝑅 ∈ V) | |
2 | dfrel2 4791 | . . 3 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅) | |
3 | cnvexg 4875 | . . . 4 ⊢ (◡𝑅 ∈ V → ◡◡𝑅 ∈ V) | |
4 | eleq1 2141 | . . . 4 ⊢ (◡◡𝑅 = 𝑅 → (◡◡𝑅 ∈ V ↔ 𝑅 ∈ V)) | |
5 | 3, 4 | syl5ib 152 | . . 3 ⊢ (◡◡𝑅 = 𝑅 → (◡𝑅 ∈ V → 𝑅 ∈ V)) |
6 | 2, 5 | sylbi 119 | . 2 ⊢ (Rel 𝑅 → (◡𝑅 ∈ V → 𝑅 ∈ V)) |
7 | 1, 6 | impbid2 141 | 1 ⊢ (Rel 𝑅 → (𝑅 ∈ V ↔ ◡𝑅 ∈ V)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 = wceq 1284 ∈ wcel 1433 Vcvv 2601 ◡ccnv 4362 Rel wrel 4368 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-xp 4369 df-rel 4370 df-cnv 4371 df-dm 4373 df-rn 4374 |
This theorem is referenced by: (None) |
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