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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 6157 | . . 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 5210 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)} |
| 4 | 1, 3 | eqtri 2765 | . 2 ⊢ ◡{〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)} |
| 5 | df-xp 5691 | . . 3 ⊢ (𝐴 × 𝐵) = {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} | |
| 6 | 5 | cnveqi 5885 | . 2 ⊢ ◡(𝐴 × 𝐵) = ◡{〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} |
| 7 | df-xp 5691 | . 2 ⊢ (𝐵 × 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)} | |
| 8 | 4, 6, 7 | 3eqtr4i 2775 | 1 ⊢ ◡(𝐴 × 𝐵) = (𝐵 × 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2108 {copab 5205 × cxp 5683 ◡ccnv 5684 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-11 2157 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 |
| This theorem is referenced by: xp0 6178 rnxp 6190 rnxpss 6192 dminxp 6200 imainrect 6201 cnvrescnv 6215 fparlem3 8139 fparlem4 8140 tposfo 8278 tposf 8279 xpider 8828 xpcomf1o 9101 fpwwe2lem12 10682 trclublem 15034 pjdm 21727 tposmap 22463 ordtrest2 23212 ustneism 24232 trust 24238 metustsym 24568 metust 24571 gtiso 32710 padct 32731 gsumhashmul 33064 ordtcnvNEW 33919 ordtrest2NEW 33922 mbfmcst 34261 eulerpartlemt 34373 0rrv 34453 msrf 35547 mthmpps 35587 elrn3 35762 trclubgNEW 43631 xpexb 44473 tposresxp 48783 tposf1o 48784 |
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