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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 5722 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴)) | |
2 | df-br 5031 | . 2 ⊢ (𝐴◡𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ ◡𝑅) | |
3 | df-br 5031 | . 2 ⊢ (𝐵𝑅𝐴 ↔ 〈𝐵, 𝐴〉 ∈ 𝑅) | |
4 | 1, 2, 3 | 3bitr3g 316 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (〈𝐴, 𝐵〉 ∈ ◡𝑅 ↔ 〈𝐵, 𝐴〉 ∈ 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2114 〈cop 4522 class class class wbr 5030 ◡ccnv 5524 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-v 3400 df-dif 3846 df-un 3848 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 df-br 5031 df-opab 5093 df-cnv 5533 |
This theorem is referenced by: opelcnv 5724 elpredim 6141 fvimacnv 6830 brtpos 7930 xrlenlt 10784 brcolinear2 33998 |
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