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Mirrors > Home > ILE Home > Th. List > opelcnvg | GIF version |
Description: Ordered-pair membership in converse. (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 | breq2 3928 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝐴)) | |
2 | breq1 3927 | . . 3 ⊢ (𝑦 = 𝐵 → (𝑦𝑅𝐴 ↔ 𝐵𝑅𝐴)) | |
3 | df-cnv 4542 | . . 3 ⊢ ◡𝑅 = {〈𝑥, 𝑦〉 ∣ 𝑦𝑅𝑥} | |
4 | 1, 2, 3 | brabg 4186 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴)) |
5 | df-br 3925 | . 2 ⊢ (𝐴◡𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ ◡𝑅) | |
6 | df-br 3925 | . 2 ⊢ (𝐵𝑅𝐴 ↔ 〈𝐵, 𝐴〉 ∈ 𝑅) | |
7 | 4, 5, 6 | 3bitr3g 221 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (〈𝐴, 𝐵〉 ∈ ◡𝑅 ↔ 〈𝐵, 𝐴〉 ∈ 𝑅)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∈ wcel 1480 〈cop 3525 class class class wbr 3924 ◡ccnv 4533 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-opab 3985 df-cnv 4542 |
This theorem is referenced by: brcnvg 4715 opelcnv 4716 fvimacnv 5528 cnvf1olem 6114 brtposg 6144 xrlenlt 7822 |
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