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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 5452 | . . 3 ⊢ ◡{〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} = {〈𝑥, 𝑦〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} | |
2 | ancom 465 | . . . 4 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) | |
3 | 2 | opabbii 4649 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)} |
4 | 1, 3 | eqtri 2632 | . 2 ⊢ ◡{〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)} |
5 | df-xp 5044 | . . 3 ⊢ (𝐴 × 𝐵) = {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} | |
6 | 5 | cnveqi 5219 | . 2 ⊢ ◡(𝐴 × 𝐵) = ◡{〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} |
7 | df-xp 5044 | . 2 ⊢ (𝐵 × 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)} | |
8 | 4, 6, 7 | 3eqtr4i 2642 | 1 ⊢ ◡(𝐴 × 𝐵) = (𝐵 × 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 ∈ wcel 1977 {copab 4642 × cxp 5036 ◡ccnv 5037 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 |
This theorem is referenced by: xp0 5471 rnxp 5483 rnxpss 5485 dminxp 5493 imainrect 5494 fparlem3 7166 fparlem4 7167 tposfo 7266 tposf 7267 xpider 7705 xpcomf1o 7934 fpwwe2lem13 9343 trclublem 13582 xpsc 16040 pjdm 19870 tposmap 20082 ordtrest2 20818 ustneism 21837 trust 21843 metustsym 22170 metust 22173 gtiso 28861 padct 28885 ordtcnvNEW 29294 ordtrest2NEW 29297 mbfmcst 29648 eulerpartlemt 29760 0rrv 29840 msrf 30693 mthmpps 30733 elrn3 30906 trclubgNEW 36944 xpexb 37679 |
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