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Mirrors > Home > ILE Home > Th. List > relbrcnvg | GIF version |
Description: When 𝑅 is a relation, the sethood assumptions on brcnv 4806 can be omitted. (Contributed by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
relbrcnvg | ⊢ (Rel 𝑅 → (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnv 5002 | . . . 4 ⊢ Rel ◡𝑅 | |
2 | brrelex12 4661 | . . . 4 ⊢ ((Rel ◡𝑅 ∧ 𝐴◡𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
3 | 1, 2 | mpan 424 | . . 3 ⊢ (𝐴◡𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
4 | 3 | a1i 9 | . 2 ⊢ (Rel 𝑅 → (𝐴◡𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))) |
5 | brrelex12 4661 | . . . 4 ⊢ ((Rel 𝑅 ∧ 𝐵𝑅𝐴) → (𝐵 ∈ V ∧ 𝐴 ∈ V)) | |
6 | 5 | ancomd 267 | . . 3 ⊢ ((Rel 𝑅 ∧ 𝐵𝑅𝐴) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
7 | 6 | ex 115 | . 2 ⊢ (Rel 𝑅 → (𝐵𝑅𝐴 → (𝐴 ∈ V ∧ 𝐵 ∈ V))) |
8 | brcnvg 4804 | . . 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 705 | 1 ⊢ (Rel 𝑅 → (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∈ wcel 2148 Vcvv 2737 class class class wbr 4000 ◡ccnv 4622 Rel wrel 4628 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-br 4001 df-opab 4062 df-xp 4629 df-rel 4630 df-cnv 4631 |
This theorem is referenced by: eliniseg2 5004 relbrcnv 5005 |
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