dmxpid - Metamath Proof Explorer

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Theorem dmxpid 5941

Description: The domain of a Cartesian square. (Contributed by NM, 28-Jul-1995.)

**Assertion**
Ref	Expression
dmxpid	⊢ dom (𝐴 × 𝐴) = 𝐴

**Proof of Theorem dmxpid**
Step	Hyp	Ref	Expression
1		dm0 5931	. . 3 ⊢ dom ∅ = ∅
2		xpeq1 5699	. . . . 5 ⊢ (𝐴 = ∅ → (𝐴 × 𝐴) = (∅ × 𝐴))
3		0xp 5784	. . . . 5 ⊢ (∅ × 𝐴) = ∅
4	2, 3	eqtrdi 2793	. . . 4 ⊢ (𝐴 = ∅ → (𝐴 × 𝐴) = ∅)
5	4	dmeqd 5916	. . 3 ⊢ (𝐴 = ∅ → dom (𝐴 × 𝐴) = dom ∅)
6		id 22	. . 3 ⊢ (𝐴 = ∅ → 𝐴 = ∅)
7	1, 5, 6	3eqtr4a 2803	. 2 ⊢ (𝐴 = ∅ → dom (𝐴 × 𝐴) = 𝐴)
8		dmxp 5939	. 2 ⊢ (𝐴 ≠ ∅ → dom (𝐴 × 𝐴) = 𝐴)
9	7, 8	pm2.61ine 3025	1 ⊢ dom (𝐴 × 𝐴) = 𝐴

Colors of variables: wff setvar class

Syntax hints: = wceq 1540 ∅c0 4333 × cxp 5683 dom cdm 5685

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-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-ne 2941 df-ral 3062 df-rex 3071 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-dm 5695

This theorem is referenced by: dmxpin 5942 xpid11 5943 sofld 6207 xpider 8828 hartogslem1 9582 unxpwdom2 9628 infxpenlem 10053 fpwwe2lem12 10682 fpwwe2 10683 canth4 10687 dmrecnq 11008 homfeqbas 17739 sscfn1 17861 sscfn2 17862 ssclem 17863 isssc 17864 rescval2 17872 issubc2 17881 cofuval 17927 resfval2 17938 resf1st 17939 psssdm2 18626 tsrss 18634 decpmatval 22771 pmatcollpw3lem 22789 ustssco 24223 ustbas2 24234 psmetdmdm 24315 xmetdmdm 24345 setsmstopn 24490 tmsval 24493 tngtopn 24671 caufval 25309 grporndm 30529 dfhnorm2 31141 hhshsslem1 31286 metideq 33892 filnetlem4 36382 poimirlem3 37630 ssbnd 37795 bnd2lem 37798 ismtyval 37807 ismndo2 37881 exidreslem 37884 divrngcl 37964 isdrngo2 37965 rtrclex 43630 fnxpdmdm 48076