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Mirrors > Home > MPE Home > Th. List > opelcnvg | Structured version Visualization version GIF version |
Description: Ordered-pair membership in converse relation. (Contributed by NM, 13-May-1999.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
opelcnvg | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (〈𝐴, 𝐵〉 ∈ ◡𝑅 ↔ 〈𝐵, 𝐴〉 ∈ 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brcnvg 5788 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴)) | |
2 | df-br 5075 | . 2 ⊢ (𝐴◡𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ ◡𝑅) | |
3 | df-br 5075 | . 2 ⊢ (𝐵𝑅𝐴 ↔ 〈𝐵, 𝐴〉 ∈ 𝑅) | |
4 | 1, 2, 3 | 3bitr3g 313 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (〈𝐴, 𝐵〉 ∈ ◡𝑅 ↔ 〈𝐵, 𝐴〉 ∈ 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∈ wcel 2106 〈cop 4567 class class class wbr 5074 ◡ccnv 5588 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-br 5075 df-opab 5137 df-cnv 5597 |
This theorem is referenced by: opelcnv 5790 fvimacnv 6930 brtpos 8051 xrlenlt 11040 brcolinear2 34360 |
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