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Mirrors > Home > MPE Home > Th. List > cnvxp | Structured version Visualization version GIF version |
Description: The converse of a Cartesian 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 6159 | . . 3 ⊢ ◡{〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} = {〈𝑥, 𝑦〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} | |
2 | ancom 460 | . . . 4 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) | |
3 | 2 | opabbii 5214 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)} |
4 | 1, 3 | eqtri 2762 | . 2 ⊢ ◡{〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)} |
5 | df-xp 5694 | . . 3 ⊢ (𝐴 × 𝐵) = {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} | |
6 | 5 | cnveqi 5887 | . 2 ⊢ ◡(𝐴 × 𝐵) = ◡{〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} |
7 | df-xp 5694 | . 2 ⊢ (𝐵 × 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)} | |
8 | 4, 6, 7 | 3eqtr4i 2772 | 1 ⊢ ◡(𝐴 × 𝐵) = (𝐵 × 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1536 ∈ wcel 2105 {copab 5209 × cxp 5686 ◡ccnv 5687 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-11 2154 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-xp 5694 df-rel 5695 df-cnv 5696 |
This theorem is referenced by: xp0 6179 rnxp 6191 rnxpss 6193 dminxp 6201 imainrect 6202 cnvrescnv 6216 fparlem3 8137 fparlem4 8138 tposfo 8276 tposf 8277 xpider 8826 xpcomf1o 9099 fpwwe2lem12 10679 trclublem 15030 pjdm 21744 tposmap 22478 ordtrest2 23227 ustneism 24247 trust 24253 metustsym 24583 metust 24586 gtiso 32715 padct 32736 gsumhashmul 33046 ordtcnvNEW 33880 ordtrest2NEW 33883 mbfmcst 34240 eulerpartlemt 34352 0rrv 34432 msrf 35526 mthmpps 35566 elrn3 35741 trclubgNEW 43607 xpexb 44449 |
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