dmxpss - Metamath Proof Explorer

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Theorem dmxpss 5995

Description: The domain of a Cartesian product is included in its first factor. (Contributed by NM, 19-Mar-2007.)

**Assertion**
Ref	Expression
dmxpss	⊢ dom (𝐴 × 𝐵) ⊆ 𝐴

**Proof of Theorem dmxpss**
Step	Hyp	Ref	Expression
1		xpeq2 5540	. . . . . 6 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅))
2		xp0 5982	. . . . . 6 ⊢ (𝐴 × ∅) = ∅
3	1, 2	eqtrdi 2849	. . . . 5 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = ∅)
4	3	dmeqd 5738	. . . 4 ⊢ (𝐵 = ∅ → dom (𝐴 × 𝐵) = dom ∅)
5		dm0 5754	. . . 4 ⊢ dom ∅ = ∅
6	4, 5	eqtrdi 2849	. . 3 ⊢ (𝐵 = ∅ → dom (𝐴 × 𝐵) = ∅)
7		0ss 4304	. . 3 ⊢ ∅ ⊆ 𝐴
8	6, 7	eqsstrdi 3969	. 2 ⊢ (𝐵 = ∅ → dom (𝐴 × 𝐵) ⊆ 𝐴)
9		dmxp 5763	. . 3 ⊢ (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
10		eqimss 3971	. . 3 ⊢ (dom (𝐴 × 𝐵) = 𝐴 → dom (𝐴 × 𝐵) ⊆ 𝐴)
11	9, 10	syl 17	. 2 ⊢ (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) ⊆ 𝐴)
12	8, 11	pm2.61ine 3070	1 ⊢ dom (𝐴 × 𝐵) ⊆ 𝐴

Colors of variables: wff setvar class

Syntax hints: = wceq 1538 ≠ wne 2987 ⊆ wss 3881 ∅c0 4243 × cxp 5517 dom cdm 5519

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-cnv 5527 df-dm 5529

This theorem is referenced by: rnxpss 5996 ssxpb 5998 resssxp 6089 funssxp 6509 dff3 6843 fparlem3 7792 fparlem4 7793 brdom3 9939 brdom5 9940 brdom4 9941 canthwelem 10061 pwfseqlem4 10073 uzrdgfni 13321 xptrrel 14331 rlimpm 14849 isohom 17038 ledm 17826 gsumxp 19089 dprd2d2 19159 tsmsxp 22760 dvbssntr 24503 gsumpart 30740 esum2d 31462 poimirlem3 35060 rtrclex 40317 trclexi 40320 rtrclexi 40321 cnvtrcl0 40326 dmtrcl 40327 rfovcnvf1od 40705 issmflem 43361