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Mirrors > Home > MPE Home > Th. List > dmxpid | Structured version Visualization version GIF version |
Description: The domain of a square Cartesian product. (Contributed by NM, 28-Jul-1995.) |
Ref | Expression |
---|---|
dmxpid | ⊢ dom (𝐴 × 𝐴) = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dm0 5260 | . . 3 ⊢ dom ∅ = ∅ | |
2 | xpeq1 5052 | . . . . 5 ⊢ (𝐴 = ∅ → (𝐴 × 𝐴) = (∅ × 𝐴)) | |
3 | 0xp 5122 | . . . . 5 ⊢ (∅ × 𝐴) = ∅ | |
4 | 2, 3 | syl6eq 2660 | . . . 4 ⊢ (𝐴 = ∅ → (𝐴 × 𝐴) = ∅) |
5 | 4 | dmeqd 5248 | . . 3 ⊢ (𝐴 = ∅ → dom (𝐴 × 𝐴) = dom ∅) |
6 | id 22 | . . 3 ⊢ (𝐴 = ∅ → 𝐴 = ∅) | |
7 | 1, 5, 6 | 3eqtr4a 2670 | . 2 ⊢ (𝐴 = ∅ → dom (𝐴 × 𝐴) = 𝐴) |
8 | dmxp 5265 | . 2 ⊢ (𝐴 ≠ ∅ → dom (𝐴 × 𝐴) = 𝐴) | |
9 | 7, 8 | pm2.61ine 2865 | 1 ⊢ dom (𝐴 × 𝐴) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∅c0 3874 × cxp 5036 dom cdm 5038 |
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-ne 2782 df-ral 2901 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-dm 5048 |
This theorem is referenced by: dmxpin 5267 xpid11 5268 sofld 5500 xpider 7705 hartogslem1 8330 unxpwdom2 8376 infxpenlem 8719 fpwwe2lem13 9343 fpwwe2 9344 canth4 9348 dmrecnq 9669 homfeqbas 16179 sscfn1 16300 sscfn2 16301 ssclem 16302 isssc 16303 rescval2 16311 issubc2 16319 cofuval 16365 resfval2 16376 resf1st 16377 psssdm2 17038 tsrss 17046 decpmatval 20389 pmatcollpw3lem 20407 ustssco 21828 ustbas2 21839 psmetdmdm 21920 xmetdmdm 21950 setsmstopn 22093 tmsval 22096 tngtopn 22264 caufval 22881 grporndm 26748 dfhnorm2 27363 hhshsslem1 27508 metideq 29264 filnetlem4 31546 poimirlem3 32582 ssbnd 32757 bnd2lem 32760 ismtyval 32769 ismndo2 32843 exidreslem 32846 divrngcl 32926 isdrngo2 32927 rtrclex 36943 fnxpdmdm 41558 |
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