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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 4898 | . . 3 ⊢ ◡{〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} = {〈𝑥, 𝑦〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} | |
2 | ancom 264 | . . . 4 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) | |
3 | 2 | opabbii 3955 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)} |
4 | 1, 3 | eqtri 2135 | . 2 ⊢ ◡{〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)} |
5 | df-xp 4505 | . . 3 ⊢ (𝐴 × 𝐵) = {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} | |
6 | 5 | cnveqi 4674 | . 2 ⊢ ◡(𝐴 × 𝐵) = ◡{〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} |
7 | df-xp 4505 | . 2 ⊢ (𝐵 × 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)} | |
8 | 4, 6, 7 | 3eqtr4i 2145 | 1 ⊢ ◡(𝐴 × 𝐵) = (𝐵 × 𝐴) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1314 ∈ wcel 1463 {copab 3948 × cxp 4497 ◡ccnv 4498 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-pr 4091 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ral 2395 df-rex 2396 df-v 2659 df-un 3041 df-in 3043 df-ss 3050 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-br 3896 df-opab 3950 df-xp 4505 df-rel 4506 df-cnv 4507 |
This theorem is referenced by: xp0 4916 rnxpm 4926 rnxpss 4928 dminxp 4941 imainrect 4942 tposfo 6122 tposf 6123 xpider 6454 xpcomf1o 6672 |
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