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Mirrors > Home > ILE Home > Th. List > cnvxp | GIF version |
Description: The converse of a cross product. Exercise 11 of [Suppes] p. 67. (Contributed by NM, 14-Aug-1999.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
cnvxp | ⊢ ◡(𝐴 × 𝐵) = (𝐵 × 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvopab 4776 | . . 3 ⊢ ◡{〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} = {〈𝑥, 𝑦〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} | |
2 | ancom 262 | . . . 4 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) | |
3 | 2 | opabbii 3865 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)} |
4 | 1, 3 | eqtri 2103 | . 2 ⊢ ◡{〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)} |
5 | df-xp 4397 | . . 3 ⊢ (𝐴 × 𝐵) = {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} | |
6 | 5 | cnveqi 4558 | . 2 ⊢ ◡(𝐴 × 𝐵) = ◡{〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} |
7 | df-xp 4397 | . 2 ⊢ (𝐵 × 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)} | |
8 | 4, 6, 7 | 3eqtr4i 2113 | 1 ⊢ ◡(𝐴 × 𝐵) = (𝐵 × 𝐴) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 = wceq 1285 ∈ wcel 1434 {copab 3858 × cxp 4389 ◡ccnv 4390 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3916 ax-pow 3968 ax-pr 3992 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ral 2358 df-rex 2359 df-v 2612 df-un 2986 df-in 2988 df-ss 2995 df-pw 3402 df-sn 3422 df-pr 3423 df-op 3425 df-br 3806 df-opab 3860 df-xp 4397 df-rel 4398 df-cnv 4399 |
This theorem is referenced by: xp0 4793 rnxpm 4802 rnxpss 4804 dminxp 4815 imainrect 4816 tposfo 5940 tposf 5941 xpiderm 6264 xpcomf1o 6390 |
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