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Mirrors > Home > ILE Home > Th. List > relbrcnvg | GIF version |
Description: When 𝑅 is a relation, the sethood assumptions on brcnv 4722 can be omitted. (Contributed by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
relbrcnvg | ⊢ (Rel 𝑅 → (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnv 4917 | . . . 4 ⊢ Rel ◡𝑅 | |
2 | brrelex12 4577 | . . . 4 ⊢ ((Rel ◡𝑅 ∧ 𝐴◡𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
3 | 1, 2 | mpan 420 | . . 3 ⊢ (𝐴◡𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
4 | 3 | a1i 9 | . 2 ⊢ (Rel 𝑅 → (𝐴◡𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))) |
5 | brrelex12 4577 | . . . 4 ⊢ ((Rel 𝑅 ∧ 𝐵𝑅𝐴) → (𝐵 ∈ V ∧ 𝐴 ∈ V)) | |
6 | 5 | ancomd 265 | . . 3 ⊢ ((Rel 𝑅 ∧ 𝐵𝑅𝐴) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
7 | 6 | ex 114 | . 2 ⊢ (Rel 𝑅 → (𝐵𝑅𝐴 → (𝐴 ∈ V ∧ 𝐵 ∈ V))) |
8 | brcnvg 4720 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴)) | |
9 | 8 | a1i 9 | . 2 ⊢ (Rel 𝑅 → ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴))) |
10 | 4, 7, 9 | pm5.21ndd 694 | 1 ⊢ (Rel 𝑅 → (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∈ wcel 1480 Vcvv 2686 class class class wbr 3929 ◡ccnv 4538 Rel wrel 4544 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-xp 4545 df-rel 4546 df-cnv 4547 |
This theorem is referenced by: relbrcnv 4919 |
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